[2282] | 1 | % ================================================================ |
---|
[3294] | 2 | % Iso-neutral diffusion : |
---|
[2282] | 3 | % ================================================================ |
---|
[3294] | 4 | \chapter[Iso-neutral diffusion and eddy advection using |
---|
| 5 | triads]{Iso-neutral diffusion and eddy advection using triads} |
---|
| 6 | \label{sec:triad} |
---|
[2282] | 7 | \minitoc |
---|
[3294] | 8 | \pagebreak |
---|
[4147] | 9 | \section{Choice of \ngn{namtra\_ldf} namelist parameters} |
---|
[3294] | 10 | %-----------------------------------------nam_traldf------------------------------------------------------ |
---|
| 11 | \namdisplay{namtra_ldf} |
---|
| 12 | %--------------------------------------------------------------------------------------------------------- |
---|
| 13 | If the namelist variable \np{ln\_traldf\_grif} is set true (and |
---|
| 14 | \key{ldfslp} is set), \NEMO updates both active and passive tracers |
---|
| 15 | using the Griffies triad representation of iso-neutral diffusion and |
---|
| 16 | the eddy-induced advective skew (GM) fluxes. Otherwise (by default) the |
---|
| 17 | filtered version of Cox's original scheme is employed |
---|
| 18 | (\S\ref{LDF_slp}). In the present implementation of the Griffies |
---|
| 19 | scheme, the advective skew fluxes are implemented even if |
---|
| 20 | \key{traldf\_eiv} is not set. |
---|
[2282] | 21 | |
---|
[3294] | 22 | Values of iso-neutral diffusivity and GM coefficient are set as |
---|
| 23 | described in \S\ref{LDF_coef}. If none of the keys \key{traldf\_cNd}, |
---|
| 24 | N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and |
---|
| 25 | GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and |
---|
| 26 | \np{rn\_aeiv\_0}. If 2D-varying coefficients are set with |
---|
| 27 | \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal |
---|
[4147] | 28 | scale factor according to \eqref{Eq_title} \footnote{Except in global ORCA |
---|
| 29 | $0.5^{\circ}$ runs with \key{traldf\_eiv}, where |
---|
[3294] | 30 | $A_l$ is set like $A_e$ but with a minimum vale of |
---|
| 31 | $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$}. In idealised setups with |
---|
| 32 | \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} |
---|
[4147] | 33 | is set in the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is |
---|
[3294] | 34 | instead set from the Held-Larichev parameterisation\footnote{In this |
---|
| 35 | case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further |
---|
| 36 | reduced by a factor $|f/f_{20}|$, where $f_{20}$ is the value of $f$ |
---|
| 37 | at $20^{\circ}$~N} (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored |
---|
| 38 | unless it is zero. |
---|
[2282] | 39 | |
---|
[3294] | 40 | The options specific to the Griffies scheme include: |
---|
| 41 | \begin{description}[font=\normalfont] |
---|
| 42 | \item[\np{ln\_traldf\_gdia}] Default value is false. See \S\ref{sec:triad:sfdiag}. If this is set true, time-mean |
---|
| 43 | eddy-advective (GM) velocities are output for diagnostic purposes, even |
---|
| 44 | though the eddy advection is accomplished by means of the skew |
---|
| 45 | fluxes. |
---|
| 46 | \item[\np{ln\_traldf\_iso}] See \S\ref{sec:triad:taper}. If this is set false (the default), then |
---|
| 47 | `iso-neutral' mixing is accomplished within the surface mixed-layer |
---|
| 48 | along slopes linearly decreasing with depth from the value immediately below |
---|
| 49 | the mixed-layer to zero (flat) at the surface (\S\ref{sec:triad:lintaper}). This is the same |
---|
| 50 | treatment as used in the default implementation |
---|
| 51 | \S\ref{LDF_slp_iso}; Fig.~\ref{Fig_eiv_slp}. Where |
---|
| 52 | \np{ln\_traldf\_iso} is set true, the vertical skew flux is further |
---|
| 53 | reduced to ensure no vertical buoyancy flux, giving an almost pure |
---|
| 54 | horizontal diffusive tracer flux within the mixed layer. This is similar to |
---|
| 55 | the tapering suggested by \citet{Gerdes1991}. See \S\ref{sec:triad:Gerdes-taper} |
---|
| 56 | \item[\np{ln\_traldf\_botmix}] See \S\ref{sec:triad:iso_bdry}. If this |
---|
| 57 | is set false (the default) then the lateral diffusive fluxes |
---|
| 58 | associated with triads partly masked by topography are neglected. If |
---|
| 59 | it is set true, however, then these lateral diffusive fluxes are |
---|
| 60 | applied, giving smoother bottom tracer fields at the cost of |
---|
| 61 | introducing diapycnal mixing. |
---|
| 62 | \end{description} |
---|
| 63 | \section{Triad formulation of iso-neutral diffusion} |
---|
| 64 | \label{sec:triad:iso} |
---|
| 65 | We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98}, but formulated within the \NEMO |
---|
[2282] | 66 | framework, using scale factors rather than grid-sizes. |
---|
| 67 | |
---|
[3294] | 68 | \subsection{The iso-neutral diffusion operator} |
---|
| 69 | The iso-neutral second order tracer diffusive operator for small |
---|
| 70 | angles between iso-neutral surfaces and geopotentials is given by |
---|
| 71 | \eqref{Eq_PE_iso_tensor}: |
---|
| 72 | \begin{subequations} \label{eq:triad:PE_iso_tensor} |
---|
| 73 | \begin{equation} |
---|
| 74 | D^{lT}=-\Div\vect{f}^{lT}\equiv |
---|
| 75 | -\frac{1}{e_1e_2e_3}\left[\pd{i}\left (f_1^{lT}e_2e_3\right) + |
---|
| 76 | \pd{j}\left (f_2^{lT}e_2e_3\right) + \pd{k}\left (f_3^{lT}e_1e_2\right)\right], |
---|
| 77 | \end{equation} |
---|
| 78 | where the diffusive flux per unit area of physical space |
---|
| 79 | \begin{equation} |
---|
| 80 | \vect{f}^{lT}=-\Alt\Re\cdot\grad T, |
---|
| 81 | \end{equation} |
---|
| 82 | \begin{equation} |
---|
| 83 | \label{eq:triad:PE_iso_tensor:c} |
---|
| 84 | \mbox{with}\quad \;\;\Re = |
---|
| 85 | \begin{pmatrix} |
---|
| 86 | 1&0&-r_1\mystrut \\ |
---|
| 87 | 0&1&-r_2\mystrut \\ |
---|
| 88 | -r_1&-r_2&r_1 ^2+r_2 ^2\mystrut |
---|
| 89 | \end{pmatrix} |
---|
| 90 | \quad \text{and} \quad\grad T= |
---|
| 91 | \begin{pmatrix} |
---|
| 92 | \frac{1}{e_1}\pd[T]{i}\mystrut \\ |
---|
| 93 | \frac{1}{e_2}\pd[T]{j}\mystrut \\ |
---|
| 94 | \frac{1}{e_3}\pd[T]{k}\mystrut |
---|
| 95 | \end{pmatrix}. |
---|
| 96 | \end{equation} |
---|
| 97 | \end{subequations} |
---|
| 98 | % \left( {{\begin{array}{*{20}c} |
---|
| 99 | % 1 \hfill & 0 \hfill & {-r_1 } \hfill \\ |
---|
| 100 | % 0 \hfill & 1 \hfill & {-r_2 } \hfill \\ |
---|
| 101 | % {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ |
---|
| 102 | % \end{array} }} \right) |
---|
| 103 | Here \eqref{Eq_PE_iso_slopes} |
---|
[2282] | 104 | \begin{align*} |
---|
| 105 | r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i} |
---|
| 106 | \right) |
---|
| 107 | \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \\ |
---|
| 108 | &=-\frac{e_3 }{e_1 } \left( -\alpha\frac{\partial T }{\partial i} + |
---|
| 109 | \beta\frac{\partial S }{\partial i} \right) \left( |
---|
| 110 | -\alpha\frac{\partial T }{\partial k} + \beta\frac{\partial S |
---|
| 111 | }{\partial k} \right)^{-1} |
---|
| 112 | \end{align*} |
---|
[3294] | 113 | is the $i$-component of the slope of the iso-neutral surface relative to the computational |
---|
| 114 | surface, and $r_2$ is the $j$-component. |
---|
[2282] | 115 | |
---|
[3294] | 116 | We will find it useful to consider the fluxes per unit area in $i,j,k$ |
---|
| 117 | space; we write |
---|
| 118 | \begin{equation} |
---|
| 119 | \label{eq:triad:Fijk} |
---|
| 120 | \vect{F}_{\mathrm{iso}}=\left(f_1^{lT}e_2e_3, f_2^{lT}e_1e_3, f_3^{lT}e_1e_2\right). |
---|
| 121 | \end{equation} |
---|
| 122 | Additionally, we will sometimes write the contributions towards the |
---|
| 123 | fluxes $\vect{f}$ and $\vect{F}_\mathrm{iso}$ from the component |
---|
| 124 | $R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$, with |
---|
| 125 | $f_{ij}=R_{ij}e_i^{-1}\partial T/\partial x_i$ (no summation) etc. |
---|
| 126 | |
---|
| 127 | The off-diagonal terms of the small angle diffusion tensor |
---|
| 128 | \eqref{Eq_PE_iso_tensor}, \eqref{eq:triad:PE_iso_tensor:c} produce skew-fluxes along the |
---|
| 129 | $i$- and $j$-directions resulting from the vertical tracer gradient: |
---|
| 130 | \begin{align} |
---|
| 131 | \label{eq:triad:i13c} |
---|
| 132 | 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}\\ |
---|
| 133 | \intertext{and in the k-direction resulting from the lateral tracer gradients} |
---|
| 134 | \label{eq:triad:i31c} |
---|
| 135 | 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} |
---|
| 136 | \end{align} |
---|
| 137 | |
---|
| 138 | The vertical diffusive flux associated with the $_{33}$ |
---|
[2282] | 139 | component of the small angle diffusion tensor is |
---|
| 140 | \begin{equation} |
---|
[3294] | 141 | \label{eq:triad:i33c} |
---|
| 142 | f_{33}=-\Alt(r_1^2 +r_2^2) \frac{1}{e_3}\frac{\partial T}{\partial k}. |
---|
[2282] | 143 | \end{equation} |
---|
| 144 | |
---|
| 145 | Since there are no cross terms involving $r_1$ and $r_2$ in the above, we can |
---|
[3294] | 146 | consider the iso-neutral diffusive fluxes separately in the $i$-$k$ and $j$-$k$ |
---|
[2282] | 147 | planes, just adding together the vertical components from each |
---|
[3294] | 148 | plane. The following description will describe the fluxes on the $i$-$k$ |
---|
[2282] | 149 | plane. |
---|
| 150 | |
---|
[3294] | 151 | There is no natural discretization for the $i$-component of the |
---|
| 152 | skew-flux, \eqref{eq:triad:i13c}, as |
---|
| 153 | although it must be evaluated at $u$-points, it involves vertical |
---|
[2282] | 154 | gradients (both for the tracer and the slope $r_1$), defined at |
---|
[3294] | 155 | $w$-points. Similarly, the vertical skew flux, \eqref{eq:triad:i31c}, is evaluated at |
---|
| 156 | $w$-points but involves horizontal gradients defined at $u$-points. |
---|
[2282] | 157 | |
---|
| 158 | \subsection{The standard discretization} |
---|
| 159 | The straightforward approach to discretize the lateral skew flux |
---|
[3294] | 160 | \eqref{eq:triad:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 |
---|
[2282] | 161 | into OPA, \eqref{Eq_tra_ldf_iso}, is to calculate a mean vertical |
---|
[3294] | 162 | gradient at the $u$-point from the average of the four surrounding |
---|
[2282] | 163 | vertical tracer gradients, and multiply this by a mean slope at the |
---|
[3294] | 164 | $u$-point, calculated from the averaged surrounding vertical density |
---|
| 165 | gradients. The total area-integrated skew-flux (flux per unit area in |
---|
| 166 | $ijk$ space) from tracer cell $i,k$ |
---|
| 167 | to $i+1,k$, noting that the $e_{{3}_{i+1/2}^k}$ in the area |
---|
| 168 | $e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with |
---|
| 169 | the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer |
---|
[2282] | 170 | gradient, is then \eqref{Eq_tra_ldf_iso} |
---|
| 171 | \begin{equation*} |
---|
[3294] | 172 | \left(F_u^{13} \right)_{i+\hhalf}^k = \Alts_{i+\hhalf}^k |
---|
| 173 | {e_{2}}_{i+1/2}^k \overline{\overline |
---|
[2282] | 174 | r_1} ^{\,i,k}\,\overline{\overline{\delta_k T}}^{\,i,k}, |
---|
| 175 | \end{equation*} |
---|
| 176 | where |
---|
| 177 | \begin{equation*} |
---|
| 178 | \overline{\overline |
---|
[3294] | 179 | r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k} |
---|
| 180 | \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}}, |
---|
[2282] | 181 | \end{equation*} |
---|
[3294] | 182 | and here and in the following we drop the $^{lT}$ superscript from |
---|
| 183 | $\Alt$ for simplicity. |
---|
[2282] | 184 | Unfortunately the resulting combination $\overline{\overline{\delta_k |
---|
| 185 | \bullet}}^{\,i,k}$ of a $k$ average and a $k$ difference %of the tracer |
---|
| 186 | reduces to $\bullet_{k+1}-\bullet_{k-1}$, so two-grid-point oscillations are |
---|
| 187 | invisible to this discretization of the iso-neutral operator. These |
---|
| 188 | \emph{computational modes} will not be damped by this operator, and |
---|
| 189 | may even possibly be amplified by it. Consequently, applying this |
---|
| 190 | operator to a tracer does not guarantee the decrease of its |
---|
| 191 | global-average variance. To correct this, we introduced a smoothing of |
---|
| 192 | the slopes of the iso-neutral surfaces (see \S\ref{LDF}). This |
---|
[3294] | 193 | technique works for $T$ and $S$ in so far as they are active tracers |
---|
[2282] | 194 | ($i.e.$ they enter the computation of density), but it does not work |
---|
| 195 | for a passive tracer. |
---|
| 196 | \subsection{Expression of the skew-flux in terms of triad slopes} |
---|
| 197 | \citep{Griffies_al_JPO98} introduce a different discretization of the |
---|
| 198 | off-diagonal terms that nicely solves the problem. |
---|
[3294] | 199 | % Instead of multiplying the mean slope calculated at the $u$-point by |
---|
| 200 | % the mean vertical gradient at the $u$-point, |
---|
[2282] | 201 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 202 | \begin{figure}[h] \begin{center} |
---|
[3297] | 203 | \includegraphics[width=1.05\textwidth]{./TexFiles/Figures/Fig_GRIFF_triad_fluxes} |
---|
[3294] | 204 | \caption{ \label{fig:triad:ISO_triad} |
---|
[2376] | 205 | (a) Arrangement of triads $S_i$ and tracer gradients to |
---|
[3294] | 206 | give lateral tracer flux from box $i,k$ to $i+1,k$ |
---|
[2376] | 207 | (b) Triads $S'_i$ and tracer gradients to give vertical tracer flux from |
---|
| 208 | box $i,k$ to $i,k+1$.} |
---|
[3294] | 209 | \end{center} \end{figure} |
---|
[2282] | 210 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 211 | They get the skew flux from the products of the vertical gradients at |
---|
[3294] | 212 | each $w$-point surrounding the $u$-point with the corresponding `triad' |
---|
| 213 | slope calculated from the lateral density gradient across the $u$-point |
---|
| 214 | divided by the vertical density gradient at the same $w$-point as the |
---|
| 215 | tracer gradient. See Fig.~\ref{fig:triad:ISO_triad}a, where the thick lines |
---|
[2282] | 216 | denote the tracer gradients, and the thin lines the corresponding |
---|
| 217 | triads, with slopes $s_1, \dotsc s_4$. The total area-integrated |
---|
| 218 | skew-flux from tracer cell $i,k$ to $i+1,k$ |
---|
| 219 | \begin{multline} |
---|
[3294] | 220 | \label{eq:triad:i13} |
---|
| 221 | \left( F_u^{13} \right)_{i+\frac{1}{2}}^k = \Alts_{i+1}^k a_1 s_1 |
---|
[2282] | 222 | \delta _{k+\frac{1}{2}} \left[ T^{i+1} |
---|
[3294] | 223 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} + \Alts _i^k a_2 s_2 \delta |
---|
[2282] | 224 | _{k+\frac{1}{2}} \left[ T^i |
---|
| 225 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} \\ |
---|
[3294] | 226 | +\Alts _{i+1}^k a_3 s_3 \delta _{k-\frac{1}{2}} \left[ T^{i+1} |
---|
| 227 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} +\Alts _i^k a_4 s_4 \delta |
---|
[2282] | 228 | _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}, |
---|
| 229 | \end{multline} |
---|
| 230 | where the contributions of the triad fluxes are weighted by areas |
---|
[3294] | 231 | $a_1, \dotsc a_4$, and $\Alts$ is now defined at the tracer points |
---|
| 232 | rather than the $u$-points. This discretization gives a much closer |
---|
[2282] | 233 | stencil, and disallows the two-point computational modes. |
---|
| 234 | |
---|
[3294] | 235 | The vertical skew flux \eqref{eq:triad:i31c} from tracer cell $i,k$ to $i,k+1$ at the |
---|
| 236 | $w$-point $i,k+\hhalf$ is constructed similarly (Fig.~\ref{fig:triad:ISO_triad}b) |
---|
[2282] | 237 | by multiplying lateral tracer gradients from each of the four |
---|
[3294] | 238 | surrounding $u$-points by the appropriate triad slope: |
---|
[2282] | 239 | \begin{multline} |
---|
[3294] | 240 | \label{eq:triad:i31} |
---|
| 241 | \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} = \Alts_i^{k+1} a_{1}' |
---|
[2282] | 242 | s_{1}' \delta _{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1} |
---|
[3294] | 243 | +\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}\\ |
---|
| 244 | + \Alts_i^k a_{3}' s_{3}' \delta _{i-\frac{1}{2}} \left[ T^k\right]/{e_{3u}}_{i-\frac{1}{2}}^k |
---|
| 245 | +\Alts_i^k a_{4}' s_{4}' \delta _{i+\frac{1}{2}} \left[ T^k \right]/{e_{3u}}_{i+\frac{1}{2}}^k. |
---|
[2282] | 246 | \end{multline} |
---|
[3294] | 247 | |
---|
| 248 | We notate the triad slopes $s_i$ and $s'_i$ in terms of the `anchor point' $i,k$ |
---|
| 249 | (appearing in both the vertical and lateral gradient), and the $u$- and |
---|
| 250 | $w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the |
---|
| 251 | triad as follows (see also Fig.~\ref{fig:triad:ISO_triad}): |
---|
[2282] | 252 | \begin{equation} |
---|
[3294] | 253 | \label{eq:triad:R} |
---|
| 254 | _i^k \mathbb{R}_{i_p}^{k_p} |
---|
| 255 | =-\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} |
---|
[2282] | 256 | \ |
---|
[3294] | 257 | \frac |
---|
[2282] | 258 | {\left(\alpha / \beta \right)_i^k \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] } |
---|
| 259 | {\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] }. |
---|
| 260 | \end{equation} |
---|
| 261 | In calculating the slopes of the local neutral |
---|
| 262 | surfaces, the expansion coefficients $\alpha$ and $\beta$ are |
---|
[3294] | 263 | evaluated at the anchor points of the triad \footnote{Note that in \eqref{eq:triad:R} we use the ratio $\alpha / \beta$ |
---|
[2282] | 264 | instead of multiplying the temperature derivative by $\alpha$ and the |
---|
| 265 | salinity derivative by $\beta$. This is more efficient as the ratio |
---|
| 266 | $\alpha / \beta$ can to be evaluated directly}, while the metrics are |
---|
[3294] | 267 | calculated at the $u$- and $w$-points on the arms. |
---|
[2282] | 268 | |
---|
| 269 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 270 | \begin{figure}[h] \begin{center} |
---|
[3297] | 271 | \includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_GRIFF_qcells} |
---|
[3294] | 272 | \caption{ \label{fig:triad:qcells} |
---|
[3297] | 273 | Triad notation for quarter cells. $T$-cells are inside |
---|
[3294] | 274 | boxes, while the $i+\half,k$ $u$-cell is shaded in green and the |
---|
| 275 | $i,k+\half$ $w$-cell is shaded in pink.} |
---|
[2282] | 276 | \end{center} \end{figure} |
---|
| 277 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 278 | |
---|
[3294] | 279 | Each triad $\{_i^k\:_{i_p}^{k_p}\}$ is associated (Fig.~\ref{fig:triad:qcells}) with the quarter |
---|
| 280 | cell that is the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ |
---|
| 281 | $u$-cell and the $i,k+k_p$ $w$-cell. Expressing the slopes $s_i$ and |
---|
| 282 | $s'_i$ in \eqref{eq:triad:i13} and \eqref{eq:triad:i31} in this notation, we have |
---|
[2282] | 283 | e.g.\ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. Each triad slope $_i^k |
---|
| 284 | \mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to calculate the |
---|
[3294] | 285 | lateral flux along its $u$-arm, at $(i+i_p,k)$, and then again as an |
---|
| 286 | $s'$ to calculate the vertical flux along its $w$-arm at |
---|
[2282] | 287 | $(i,k+k_p)$. Each vertical area $a_i$ used to calculate the lateral |
---|
| 288 | flux and horizontal area $a'_i$ used to calculate the vertical flux |
---|
[3294] | 289 | can also be identified as the area across the $u$- and $w$-arms of a |
---|
| 290 | unique triad, and we notate these areas, similarly to the triad |
---|
[2282] | 291 | slopes, as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, |
---|
[3294] | 292 | $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, where e.g. in \eqref{eq:triad:i13} |
---|
| 293 | $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, and in \eqref{eq:triad:i31} |
---|
[2282] | 294 | $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. |
---|
| 295 | |
---|
| 296 | \subsection{The full triad fluxes} |
---|
[3294] | 297 | A key property of iso-neutral diffusion is that it should not affect |
---|
[2282] | 298 | the (locally referenced) density. In particular there should be no |
---|
| 299 | lateral or vertical density flux. The lateral density flux disappears so long as the |
---|
| 300 | area-integrated lateral diffusive flux from tracer cell $i,k$ to |
---|
| 301 | $i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the |
---|
| 302 | form |
---|
| 303 | \begin{equation} |
---|
[3294] | 304 | \label{eq:triad:i11} |
---|
[2282] | 305 | \left( F_u^{11} \right) _{i+\frac{1}{2}} ^{k} = |
---|
[3294] | 306 | - \left( \Alts_i^{k+1} a_{1} + \Alts_i^{k+1} a_{2} + \Alts_i^k |
---|
| 307 | a_{3} + \Alts_i^k a_{4} \right) |
---|
[2282] | 308 | \frac{\delta _{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}}, |
---|
| 309 | \end{equation} |
---|
[3294] | 310 | where the areas $a_i$ are as in \eqref{eq:triad:i13}. In this case, |
---|
| 311 | separating the total lateral flux, the sum of \eqref{eq:triad:i13} and |
---|
| 312 | \eqref{eq:triad:i11}, into triad components, a lateral tracer |
---|
[2282] | 313 | flux |
---|
| 314 | \begin{equation} |
---|
[3294] | 315 | \label{eq:triad:latflux-triad} |
---|
| 316 | _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) = - \Alts_i^k{ \:}_i^k{\mathbb{A}_u}_{i_p}^{k_p} |
---|
[2282] | 317 | \left( |
---|
| 318 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 319 | -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
---|
| 320 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
| 321 | \right) |
---|
| 322 | \end{equation} |
---|
| 323 | can be identified with each triad. Then, because the |
---|
| 324 | same metric factors ${e_{3w}}_{\,i}^{\,k+k_p}$ and |
---|
| 325 | ${e_{1u}}_{\,i+i_p}^{\,k}$ are employed for both the density gradients |
---|
| 326 | in $ _i^k \mathbb{R}_{i_p}^{k_p}$ and the tracer gradients, the lateral |
---|
| 327 | density flux associated with each triad separately disappears. |
---|
| 328 | \begin{equation} |
---|
[3294] | 329 | \label{eq:triad:latflux-rho} |
---|
[2282] | 330 | {\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 |
---|
| 331 | \end{equation} |
---|
| 332 | Thus the total flux $\left( F_u^{31} \right) ^i _{i,k+\frac{1}{2}} + |
---|
| 333 | \left( F_u^{11} \right) ^i _{i,k+\frac{1}{2}}$ from tracer cell $i,k$ |
---|
| 334 | to $i+1,k$ must also vanish since it is a sum of four such triad fluxes. |
---|
| 335 | |
---|
[3294] | 336 | The squared slope $r_1^2$ in the expression \eqref{eq:triad:i33c} for the |
---|
[2282] | 337 | $_{33}$ component is also expressed in terms of area-weighted |
---|
| 338 | squared triad slopes, so the area-integrated vertical flux from tracer |
---|
| 339 | cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is |
---|
| 340 | \begin{equation} |
---|
[3294] | 341 | \label{eq:triad:i33} |
---|
[2282] | 342 | \left( F_w^{33} \right) _i^{k+\frac{1}{2}} = |
---|
[3294] | 343 | - \left( \Alts_i^{k+1} a_{1}' s_{1}'^2 |
---|
| 344 | + \Alts_i^{k+1} a_{2}' s_{2}'^2 |
---|
| 345 | + \Alts_i^k a_{3}' s_{3}'^2 |
---|
| 346 | + \Alts_i^k a_{4}' s_{4}'^2 \right)\delta_{k+\frac{1}{2}} \left[ T^{i+1} \right], |
---|
[2282] | 347 | \end{equation} |
---|
| 348 | where the areas $a'$ and slopes $s'$ are the same as in |
---|
[3294] | 349 | \eqref{eq:triad:i31}. |
---|
| 350 | Then, separating the total vertical flux, the sum of \eqref{eq:triad:i31} and |
---|
| 351 | \eqref{eq:triad:i33}, into triad components, a vertical flux |
---|
[2282] | 352 | \begin{align} |
---|
[3294] | 353 | \label{eq:triad:vertflux-triad} |
---|
[2282] | 354 | _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) |
---|
[3294] | 355 | &= \Alts_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p} |
---|
[2282] | 356 | \left( |
---|
| 357 | {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 358 | -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ |
---|
| 359 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
| 360 | \right) \\ |
---|
| 361 | &= - \left(\left.{ }_i^k{\mathbb{A}_w}_{i_p}^{k_p}\right/{ }_i^k{\mathbb{A}_u}_{i_p}^{k_p}\right) |
---|
[3294] | 362 | {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:triad:vertflux-triad2} |
---|
[2282] | 363 | \end{align} |
---|
| 364 | may be associated with each triad. Each vertical density flux $_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)$ |
---|
| 365 | associated with a triad then separately disappears (because the |
---|
| 366 | lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (\rho)$ |
---|
| 367 | disappears). Consequently the total vertical density flux $\left( F_w^{31} \right)_i ^{k+\frac{1}{2}} + |
---|
| 368 | \left( F_w^{33} \right)_i^{k+\frac{1}{2}}$ from tracer cell $i,k$ |
---|
| 369 | to $i,k+1$ must also vanish since it is a sum of four such triad |
---|
| 370 | fluxes. |
---|
| 371 | |
---|
[3294] | 372 | 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 |
---|
| 373 | the $u$-fluxes and $w$-fluxes in |
---|
| 374 | \eqref{eq:triad:i31}, \eqref{eq:triad:i13}, \eqref{eq:triad:i11} \eqref{eq:triad:i33} and |
---|
| 375 | Fig.~\ref{fig:triad:ISO_triad} to write out the iso-neutral fluxes at $u$- and |
---|
[2282] | 376 | $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces: |
---|
| 377 | %(Fig.~\ref{Fig_ISO_triad}): |
---|
[3294] | 378 | \begin{flalign} \label{Eq_iso_flux} \vect{F}_\mathrm{iso}(T) &\equiv |
---|
[2282] | 379 | \sum_{\substack{i_p,\,k_p}} |
---|
| 380 | \begin{pmatrix} |
---|
| 381 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ |
---|
| 382 | \\ |
---|
| 383 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) \\ |
---|
| 384 | \end{pmatrix}. |
---|
| 385 | \end{flalign} |
---|
[3294] | 386 | \subsection{Ensuring the scheme does not increase tracer variance} |
---|
| 387 | \label{sec:triad:variance} |
---|
[2282] | 388 | |
---|
[3294] | 389 | We now require that this operator should not increase the |
---|
[2282] | 390 | globally-integrated tracer variance. |
---|
| 391 | %This changes according to |
---|
| 392 | % \begin{align*} |
---|
| 393 | % &\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\ |
---|
[3294] | 394 | % &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
| 395 | % \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
---|
[2282] | 396 | % + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\ |
---|
[3294] | 397 | % &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
[2282] | 398 | % {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] |
---|
| 399 | % + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\ |
---|
| 400 | % \end{align*} |
---|
| 401 | Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ drives a lateral flux |
---|
[3294] | 402 | $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ across the $u$-point $i+i_p,k$ and |
---|
[2282] | 403 | a vertical flux $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ across the |
---|
[3294] | 404 | $w$-point $i,k+k_p$. The lateral flux drives a net rate of change of |
---|
| 405 | variance, summed over the two $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of |
---|
[2282] | 406 | \begin{multline} |
---|
| 407 | {b_T}_{i+i_p-1/2}^k\left(\frac{\partial T}{\partial t}T\right)_{i+i_p-1/2}^k+ |
---|
| 408 | \quad {b_T}_{i+i_p+1/2}^k\left(\frac{\partial T}{\partial |
---|
| 409 | t}T\right)_{i+i_p+1/2}^k \\ |
---|
| 410 | \begin{split} |
---|
| 411 | &= -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 |
---|
| 412 | {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\ |
---|
[3294] | 413 | &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:triad:dvar_iso_i} |
---|
[2282] | 414 | \end{split} |
---|
| 415 | \end{multline} |
---|
[3294] | 416 | while the vertical flux similarly drives a net rate of change of |
---|
| 417 | variance summed over the $T$-points $i,k+k_p-\half$ (above) and |
---|
| 418 | $i,k+k_p+\half$ (below) of |
---|
[2282] | 419 | \begin{equation} |
---|
[3294] | 420 | \label{eq:triad:dvar_iso_k} |
---|
[2282] | 421 | _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. |
---|
| 422 | \end{equation} |
---|
| 423 | The total variance tendency driven by the triad is the sum of these |
---|
| 424 | two. Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and |
---|
[3294] | 425 | $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with \eqref{eq:triad:latflux-triad} and |
---|
| 426 | \eqref{eq:triad:vertflux-triad}, it is |
---|
[2282] | 427 | \begin{multline*} |
---|
[3294] | 428 | -\Alts_i^k\left \{ |
---|
[2282] | 429 | { } _i^k{\mathbb{A}_u}_{i_p}^{k_p} |
---|
| 430 | \left( |
---|
| 431 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 432 | - {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
---|
| 433 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }\right)\,\delta_{i+ i_p}[T^k] \right.\\ |
---|
| 434 | - \left. { } _i^k{\mathbb{A}_w}_{i_p}^{k_p} |
---|
| 435 | \left( |
---|
| 436 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 437 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
| 438 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
| 439 | \right) {\,}_i^k\mathbb{R}_{i_p}^{k_p}\delta_{k+ k_p}[T^i] |
---|
| 440 | \right \}. |
---|
| 441 | \end{multline*} |
---|
| 442 | The key point is then that if we require |
---|
| 443 | $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$ and $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$ |
---|
| 444 | to be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by |
---|
| 445 | \begin{equation} |
---|
[3294] | 446 | \label{eq:triad:V-A} |
---|
[2282] | 447 | _i^k\mathbb{V}_{i_p}^{k_p} |
---|
| 448 | ={\;}_i^k{\mathbb{A}_u}_{i_p}^{k_p}\,{e_{1u}}_{\,i + i_p}^{\,k} |
---|
| 449 | ={\;}_i^k{\mathbb{A}_w}_{i_p}^{k_p}\,{e_{3w}}_{\,i}^{\,k + k_p}, |
---|
| 450 | \end{equation} |
---|
| 451 | the variance tendency reduces to the perfect square |
---|
| 452 | \begin{equation} |
---|
[3294] | 453 | \label{eq:triad:perfect-square} |
---|
| 454 | -\Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} |
---|
[2282] | 455 | \left( |
---|
| 456 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 457 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
| 458 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
| 459 | \right)^2\leq 0. |
---|
| 460 | \end{equation} |
---|
[3294] | 461 | Thus, the constraint \eqref{eq:triad:V-A} ensures that the fluxes (\ref{eq:triad:latflux-triad}, \ref{eq:triad:vertflux-triad}) associated |
---|
[2282] | 462 | with a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase |
---|
| 463 | the net variance. Since the total fluxes are sums of such fluxes from |
---|
| 464 | the various triads, this constraint, applied to all triads, is |
---|
| 465 | sufficient to ensure that the globally integrated variance does not |
---|
| 466 | increase. |
---|
| 467 | |
---|
[3294] | 468 | The expression \eqref{eq:triad:V-A} can be interpreted as a discretization |
---|
[2282] | 469 | of the global integral |
---|
| 470 | \begin{equation} |
---|
[3294] | 471 | \label{eq:triad:cts-var} |
---|
| 472 | \frac{\partial}{\partial t}\int\!\half T^2\, dV = |
---|
| 473 | \int\!\mathbf{F}\cdot\nabla T\, dV, |
---|
[2282] | 474 | \end{equation} |
---|
| 475 | where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the |
---|
| 476 | lateral and vertical fluxes/unit area |
---|
| 477 | \[ |
---|
| 478 | \mathbf{F}=\left( |
---|
[3294] | 479 | \left.{}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_u}_{i_p}^{k_p}, |
---|
| 480 | \left.{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_w}_{i_p}^{k_p} |
---|
[2282] | 481 | \right) |
---|
| 482 | \] |
---|
| 483 | and the gradient |
---|
| 484 | \[\nabla T = \left( |
---|
[3294] | 485 | \left.\delta_{i+ i_p}[T^k] \right/ {e_{1u}}_{\,i + i_p}^{\,k}, |
---|
| 486 | \left.\delta_{k+ k_p}[T^i] \right/ {e_{3w}}_{\,i}^{\,k + k_p} |
---|
[2282] | 487 | \right) |
---|
| 488 | \] |
---|
| 489 | \subsection{Triad volumes in Griffes's scheme and in \NEMO} |
---|
| 490 | To complete the discretization we now need only specify the triad |
---|
| 491 | volumes $_i^k\mathbb{V}_{i_p}^{k_p}$. \citet{Griffies_al_JPO98} identify |
---|
| 492 | these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter |
---|
[3294] | 493 | cells, defined in terms of the distances between $T$, $u$,$f$ and |
---|
| 494 | $w$-points. This is the natural discretization of |
---|
| 495 | \eqref{eq:triad:cts-var}. The \NEMO model, however, operates with scale |
---|
[2282] | 496 | factors instead of grid sizes, and scale factors for the quarter |
---|
| 497 | cells are not defined. Instead, therefore we simply choose |
---|
| 498 | \begin{equation} |
---|
[3294] | 499 | \label{eq:triad:V-NEMO} |
---|
[2282] | 500 | _i^k\mathbb{V}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k, |
---|
| 501 | \end{equation} |
---|
[3294] | 502 | as a quarter of the volume of the $u$-cell inside which the triad |
---|
[2282] | 503 | quarter-cell lies. This has the nice property that when the slopes |
---|
| 504 | $\mathbb{R}$ vanish, the lateral flux from tracer cell $i,k$ to |
---|
| 505 | $i+1,k$ reduces to the classical form |
---|
| 506 | \begin{equation} |
---|
[3294] | 507 | \label{eq:triad:lat-normal} |
---|
| 508 | -\overline\Alts_{\,i+1/2}^k\; |
---|
[2282] | 509 | \frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
| 510 | \;\frac{\delta_{i+ 1/2}[T^k] }{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
[3294] | 511 | = -\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}}. |
---|
[2282] | 512 | \end{equation} |
---|
[3294] | 513 | In fact if the diffusive coefficient is defined at $u$-points, so that |
---|
| 514 | we employ $\Alts_{i+i_p}^k$ instead of $\Alts_i^k$ in the definitions of the |
---|
| 515 | triad fluxes \eqref{eq:triad:latflux-triad} and \eqref{eq:triad:vertflux-triad}, |
---|
[2282] | 516 | we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above. |
---|
| 517 | |
---|
| 518 | \subsection{Summary of the scheme} |
---|
[3294] | 519 | The iso-neutral fluxes at $u$- and |
---|
| 520 | $w$-points are the sums of the triad fluxes that cross the $u$- and |
---|
| 521 | $w$-faces \eqref{Eq_iso_flux}: |
---|
| 522 | \begin{subequations}\label{eq:triad:alltriadflux} |
---|
| 523 | \begin{flalign}\label{eq:triad:vect_isoflux} |
---|
| 524 | \vect{F}_\mathrm{iso}(T) &\equiv |
---|
| 525 | \sum_{\substack{i_p,\,k_p}} |
---|
| 526 | \begin{pmatrix} |
---|
| 527 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ |
---|
| 528 | \\ |
---|
| 529 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
---|
| 530 | \end{pmatrix}, |
---|
| 531 | \end{flalign} |
---|
| 532 | where \eqref{eq:triad:latflux-triad}: |
---|
| 533 | \begin{align} |
---|
| 534 | \label{eq:triad:triadfluxu} |
---|
| 535 | _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - \Alts_i^k{ |
---|
| 536 | \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
| 537 | \left( |
---|
| 538 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 539 | -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
---|
| 540 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
| 541 | \right),\\ |
---|
| 542 | \intertext{and} |
---|
| 543 | _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) |
---|
| 544 | &= \Alts_i^k{\: }\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{3w}}_{\,i}^{\,k+k_p}} |
---|
| 545 | \left( |
---|
| 546 | {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 547 | -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ |
---|
| 548 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
| 549 | \right),\label{eq:triad:triadfluxw} |
---|
| 550 | \end{align} |
---|
| 551 | with \eqref{eq:triad:V-NEMO} |
---|
| 552 | \begin{equation} |
---|
| 553 | \label{eq:triad:V-NEMO2} |
---|
| 554 | _i^k{\mathbb{V}}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k. |
---|
| 555 | \end{equation} |
---|
| 556 | \end{subequations} |
---|
| 557 | |
---|
| 558 | The divergence of the expression \eqref{Eq_iso_flux} for the fluxes gives the iso-neutral diffusion tendency at |
---|
[2282] | 559 | each tracer point: |
---|
[3294] | 560 | \begin{equation} \label{eq:triad:iso_operator} D_l^T = \frac{1}{b_T} |
---|
[2282] | 561 | \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k |
---|
| 562 | {\mathbb{F}_u }_{i_p}^{k_p}} \right] + \delta_{k} \left[ |
---|
| 563 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} |
---|
| 564 | \end{equation} |
---|
| 565 | where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. |
---|
| 566 | The diffusion scheme satisfies the following six properties: |
---|
| 567 | \begin{description} |
---|
| 568 | \item[$\bullet$ horizontal diffusion] The discretization of the |
---|
[3294] | 569 | diffusion operator recovers \eqref{eq:triad:lat-normal} the traditional five-point Laplacian in |
---|
[2282] | 570 | the limit of flat iso-neutral direction : |
---|
[3294] | 571 | \begin{equation} \label{eq:triad:iso_property0} D_l^T = \frac{1}{b_T} \ |
---|
[2282] | 572 | \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; |
---|
[3294] | 573 | \overline\Alts^{\,i} \; \delta_{i+1/2}[T] \right] \qquad |
---|
[2282] | 574 | \text{when} \quad { _i^k \mathbb{R}_{i_p}^{k_p} }=0 |
---|
| 575 | \end{equation} |
---|
| 576 | |
---|
| 577 | \item[$\bullet$ implicit treatment in the vertical] Only tracer values |
---|
| 578 | associated with a single water column appear in the expression |
---|
[3294] | 579 | \eqref{eq:triad:i33} for the $_{33}$ fluxes, vertical fluxes driven by |
---|
[2282] | 580 | vertical gradients. This is of paramount importance since it means |
---|
[3294] | 581 | that a time-implicit algorithm can be used to solve the vertical |
---|
| 582 | diffusion equation. This is necessary |
---|
| 583 | since the vertical eddy |
---|
[2282] | 584 | diffusivity associated with this term, |
---|
| 585 | \begin{equation} |
---|
[3294] | 586 | \frac{1}{b_w}\sum_{\substack{i_p, \,k_p}} \left\{ |
---|
| 587 | {\:}_i^k\mathbb{V}_{i_p}^{k_p} \: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
---|
| 588 | \right\} = |
---|
| 589 | \frac{1}{4b_w}\sum_{\substack{i_p, \,k_p}} \left\{ |
---|
| 590 | {b_u}_{i+i_p}^k\: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
---|
[2282] | 591 | \right\}, |
---|
| 592 | \end{equation} |
---|
| 593 | (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large. |
---|
| 594 | |
---|
| 595 | \item[$\bullet$ pure iso-neutral operator] The iso-neutral flux of |
---|
| 596 | locally referenced potential density is zero. See |
---|
[3294] | 597 | \eqref{eq:triad:latflux-rho} and \eqref{eq:triad:vertflux-triad2}. |
---|
[2282] | 598 | |
---|
| 599 | \item[$\bullet$ conservation of tracer] The iso-neutral diffusion |
---|
| 600 | conserves tracer content, $i.e.$ |
---|
[3294] | 601 | \begin{equation} \label{eq:triad:iso_property1} \sum_{i,j,k} \left\{ D_l^T \ |
---|
[2282] | 602 | b_T \right\} = 0 |
---|
| 603 | \end{equation} |
---|
| 604 | This property is trivially satisfied since the iso-neutral diffusive |
---|
| 605 | operator is written in flux form. |
---|
| 606 | |
---|
| 607 | \item[$\bullet$ no increase of tracer variance] The iso-neutral diffusion |
---|
| 608 | does not increase the tracer variance, $i.e.$ |
---|
[3294] | 609 | \begin{equation} \label{eq:triad:iso_property2} \sum_{i,j,k} \left\{ T \ D_l^T |
---|
[2282] | 610 | \ b_T \right\} \leq 0 |
---|
| 611 | \end{equation} |
---|
| 612 | The property is demonstrated in |
---|
[3294] | 613 | \S\ref{sec:triad:variance} above. It is a key property for a diffusion |
---|
| 614 | term. It means that it is also a dissipation term, $i.e.$ it |
---|
| 615 | dissipates the square of the quantity on which it is applied. It |
---|
[2282] | 616 | therefore ensures that, when the diffusivity coefficient is large |
---|
[3294] | 617 | enough, the field on which it is applied becomes free of grid-point |
---|
[2282] | 618 | noise. |
---|
| 619 | |
---|
| 620 | \item[$\bullet$ self-adjoint operator] The iso-neutral diffusion |
---|
| 621 | operator is self-adjoint, $i.e.$ |
---|
[3294] | 622 | \begin{equation} \label{eq:triad:iso_property3} \sum_{i,j,k} \left\{ S \ D_l^T |
---|
[2282] | 623 | \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} |
---|
| 624 | \end{equation} |
---|
| 625 | In other word, there is no need to develop a specific routine from |
---|
| 626 | the adjoint of this operator. We just have to apply the same |
---|
| 627 | routine. This property can be demonstrated similarly to the proof of |
---|
| 628 | the `no increase of tracer variance' property. The contribution by a |
---|
[3294] | 629 | single triad towards the left hand side of \eqref{eq:triad:iso_property3}, can |
---|
| 630 | be found by replacing $\delta[T]$ by $\delta[S]$ in \eqref{eq:triad:dvar_iso_i} |
---|
| 631 | and \eqref{eq:triad:dvar_iso_k}. This results in a term similar to |
---|
| 632 | \eqref{eq:triad:perfect-square}, |
---|
[2282] | 633 | \begin{equation} |
---|
[3294] | 634 | \label{eq:triad:TScovar} |
---|
| 635 | - \Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} |
---|
[2282] | 636 | \left( |
---|
| 637 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 638 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
| 639 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
| 640 | \right) |
---|
| 641 | \left( |
---|
| 642 | \frac{ \delta_{i+ i_p}[S^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 643 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
| 644 | \frac{ \delta_{k+k_p} [S^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
| 645 | \right). |
---|
| 646 | \end{equation} |
---|
| 647 | This is symmetrical in $T $ and $S$, so exactly the same term arises |
---|
| 648 | from the discretization of this triad's contribution towards the |
---|
[3294] | 649 | RHS of \eqref{eq:triad:iso_property3}. |
---|
[2282] | 650 | \end{description} |
---|
[3294] | 651 | \subsection{Treatment of the triads at the boundaries}\label{sec:triad:iso_bdry} |
---|
| 652 | The triad slope can only be defined where both the grid boxes centred at |
---|
| 653 | the end of the arms exist. Triads that would poke up |
---|
| 654 | through the upper ocean surface into the atmosphere, or down into the |
---|
| 655 | ocean floor, must be masked out. See Fig.~\ref{fig:triad:bdry_triads}. Surface layer triads |
---|
| 656 | $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and |
---|
| 657 | $\triad{i+1}{1}{R}{-1/2}{-1/2}$ (blue) that require density to be |
---|
| 658 | specified above the ocean surface are masked (Fig.~\ref{fig:triad:bdry_triads}a): this ensures that lateral |
---|
| 659 | tracer gradients produce no flux through the ocean surface. However, |
---|
| 660 | to prevent surface noise, it is customary to retain the $_{11}$ contributions towards |
---|
| 661 | the lateral triad fluxes $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and |
---|
| 662 | $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$; this drives diapycnal tracer |
---|
| 663 | fluxes. Similar comments apply to triads that would intersect the |
---|
| 664 | ocean floor (Fig.~\ref{fig:triad:bdry_triads}b). Note that both near bottom |
---|
| 665 | triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and |
---|
| 666 | $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ |
---|
| 667 | or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point is |
---|
| 668 | masked. The associated lateral fluxes (grey-black dashed line) are |
---|
| 669 | masked if \np{ln\_botmix\_grif}=false, but left unmasked, |
---|
| 670 | giving bottom mixing, if \np{ln\_botmix\_grif}=true. |
---|
[2282] | 671 | |
---|
[3294] | 672 | The default option \np{ln\_botmix\_grif}=false is suitable when the |
---|
| 673 | bbl mixing option is enabled (\key{trabbl}, with \np{nn\_bbl\_ldf}=1), |
---|
| 674 | or for simple idealized problems. For setups with topography without |
---|
| 675 | bbl mixing, \np{ln\_botmix\_grif}=true may be necessary. |
---|
| 676 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 677 | \begin{figure}[h] \begin{center} |
---|
[3297] | 678 | \includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_GRIFF_bdry_triads} |
---|
[3294] | 679 | \caption{ \label{fig:triad:bdry_triads} |
---|
| 680 | (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer |
---|
| 681 | points (black dots), and $i+1/2,1$ $u$-point (blue square). Triad |
---|
| 682 | slopes $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and $\triad{i+1}{1}{R}{-1/2}{-1/2}$ |
---|
| 683 | (blue) poking through the ocean surface are masked (faded in |
---|
| 684 | figure). However, the lateral $_{11}$ contributions towards |
---|
| 685 | $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$ |
---|
| 686 | (yellow line) are still applied, giving diapycnal diffusive |
---|
| 687 | fluxes.\\ |
---|
| 688 | (b) Both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and |
---|
| 689 | $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ |
---|
| 690 | or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point |
---|
| 691 | is masked. The associated lateral fluxes (grey-black dashed |
---|
| 692 | line) are masked if \np{botmix\_grif}=.false., but left |
---|
| 693 | unmasked, giving bottom mixing, if \np{botmix\_grif}=.true.} |
---|
| 694 | \end{center} \end{figure} |
---|
| 695 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 696 | \subsection{ Limiting of the slopes within the interior}\label{sec:triad:limit} |
---|
| 697 | As discussed in \S\ref{LDF_slp_iso}, iso-neutral slopes relative to |
---|
| 698 | geopotentials must be bounded everywhere, both for consistency with the small-slope |
---|
| 699 | approximation and for numerical stability \citep{Cox1987, |
---|
| 700 | Griffies_Bk04}. The bound chosen in \NEMO is applied to each |
---|
| 701 | component of the slope separately and has a value of $1/100$ in the ocean interior. |
---|
| 702 | %, ramping linearly down above 70~m depth to zero at the surface |
---|
| 703 | It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to |
---|
| 704 | geopotentials (here the $\sigma_i$ are the slopes of the coordinate surfaces relative to |
---|
| 705 | geopotentials) \eqref{Eq_PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate |
---|
| 706 | surfaces, so we require |
---|
| 707 | \begin{equation*} |
---|
| 708 | |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01. |
---|
| 709 | \end{equation*} |
---|
| 710 | and then recalculate the slopes $r_i$ relative to coordinates. |
---|
| 711 | Each individual triad slope |
---|
| 712 | \begin{equation} |
---|
| 713 | \label{eq:triad:Rtilde} |
---|
| 714 | _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}} |
---|
| 715 | \end{equation} |
---|
| 716 | is limited like this and then the corresponding |
---|
| 717 | $_i^k\mathbb{R}_{i_p}^{k_p} $ are recalculated and combined to form the fluxes. |
---|
| 718 | Note that where the slopes have been limited, there is now a non-zero |
---|
| 719 | iso-neutral density flux that drives dianeutral mixing. In particular this iso-neutral density flux |
---|
| 720 | is always downwards, and so acts to reduce gravitational potential energy. |
---|
| 721 | \subsection{Tapering within the surface mixed layer}\label{sec:triad:taper} |
---|
[2282] | 722 | |
---|
[3294] | 723 | Additional tapering of the iso-neutral fluxes is necessary within the |
---|
| 724 | surface mixed layer. When the Griffies triads are used, we offer two |
---|
| 725 | options for this. |
---|
| 726 | \subsubsection{Linear slope tapering within the surface mixed layer}\label{sec:triad:lintaper} |
---|
| 727 | This is the option activated by the default choice |
---|
| 728 | \np{ln\_triad\_iso}=false. Slopes $\tilde{r}_i$ relative to |
---|
| 729 | geopotentials are tapered linearly from their value immediately below the mixed layer to zero at the |
---|
| 730 | surface, as described in option (c) of Fig.~\ref{Fig_eiv_slp}, to values |
---|
| 731 | \begin{subequations} |
---|
| 732 | \begin{equation} |
---|
| 733 | \label{eq:triad:rmtilde} |
---|
| 734 | \rMLt = |
---|
| 735 | -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for } z>-h, |
---|
| 736 | \end{equation} |
---|
| 737 | and then the $r_i$ relative to vertical coordinate surfaces are appropriately |
---|
| 738 | adjusted to |
---|
| 739 | \begin{equation} |
---|
| 740 | \label{eq:triad:rm} |
---|
| 741 | \rML =\rMLt -\sigma_i \quad \text{ for } z>-h. |
---|
| 742 | \end{equation} |
---|
| 743 | \end{subequations} |
---|
| 744 | Thus the diffusion operator within the mixed layer is given by: |
---|
| 745 | \begin{equation} \label{eq:triad:iso_tensor_ML} |
---|
| 746 | D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad |
---|
| 747 | \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} |
---|
| 748 | 1 \hfill & 0 \hfill & {-\rML[1]}\hfill \\ |
---|
| 749 | 0 \hfill & 1 \hfill & {-\rML[2]} \hfill \\ |
---|
| 750 | {-\rML[1]}\hfill & {-\rML[2]} \hfill & {\rML[1]^2+\rML[2]^2} \hfill |
---|
| 751 | \end{array} }} \right) |
---|
| 752 | \end{equation} |
---|
| 753 | |
---|
| 754 | This slope tapering gives a natural connection between tracer in the |
---|
| 755 | mixed-layer and in isopycnal layers immediately below, in the |
---|
| 756 | thermocline. It is consistent with the way the $\tilde{r}_i$ are |
---|
| 757 | tapered within the mixed layer (see \S\ref{sec:triad:taperskew} below) |
---|
| 758 | so as to ensure a uniform GM eddy-induced velocity throughout the |
---|
| 759 | mixed layer. However, it gives a downwards density flux and so acts so |
---|
| 760 | as to reduce potential energy in the same way as does the slope |
---|
| 761 | limiting discussed above in \S\ref{sec:triad:limit}. |
---|
| 762 | |
---|
| 763 | As in \S\ref{sec:triad:limit} above, the tapering |
---|
| 764 | \eqref{eq:triad:rmtilde} is applied separately to each triad |
---|
| 765 | $_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the |
---|
| 766 | $_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted. For clarity, we assume |
---|
| 767 | $z$-coordinates in the following; the conversion from |
---|
| 768 | $\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as described |
---|
| 769 | above by \eqref{eq:triad:Rtilde}. |
---|
| 770 | \begin{enumerate} |
---|
| 771 | \item Mixed-layer depth is defined so as to avoid including regions of weak |
---|
| 772 | vertical stratification in the slope definition. |
---|
| 773 | At each $i,j$ (simplified to $i$ in |
---|
| 774 | Fig.~\ref{fig:triad:MLB_triad}), we define the mixed-layer by setting |
---|
| 775 | the vertical index of the tracer point immediately below the mixed |
---|
| 776 | layer, $k_\mathrm{ML}$, as the maximum $k$ (shallowest tracer point) |
---|
| 777 | such that the potential density |
---|
| 778 | ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, where $i,k_{10}$ is |
---|
| 779 | the tracer gridbox within which the depth reaches 10~m. See the left |
---|
| 780 | side of Fig.~\ref{fig:triad:MLB_triad}. We use the $k_{10}$-gridbox |
---|
| 781 | instead of the surface gridbox to avoid problems e.g.\ with thin |
---|
| 782 | daytime mixed-layers. Currently we use the same |
---|
| 783 | $\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is |
---|
| 784 | used to output the diagnosed mixed-layer depth |
---|
| 785 | $h_\mathrm{ML}=|z_{W}|_{k_\mathrm{ML}+1/2}$, the depth of the $w$-point |
---|
| 786 | above the $i,k_\mathrm{ML}$ tracer point. |
---|
| 787 | |
---|
| 788 | \item We define `basal' triad slopes |
---|
| 789 | ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ as the slopes |
---|
| 790 | of those triads whose vertical `arms' go down from the |
---|
| 791 | $i,k_\mathrm{ML}$ tracer point to the $i,k_\mathrm{ML}-1$ tracer point |
---|
| 792 | below. This is to ensure that the vertical density gradients |
---|
| 793 | associated with these basal triad slopes |
---|
| 794 | ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ are |
---|
| 795 | representative of the thermocline. The four basal triads defined in the bottom part |
---|
| 796 | of Fig.~\ref{fig:triad:MLB_triad} are then |
---|
| 797 | \begin{align} |
---|
| 798 | {\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p} &= |
---|
| 799 | {\:}^{k_\mathrm{ML}-k_p-1/2}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}, \label{eq:triad:Rbase} |
---|
| 800 | \\ |
---|
| 801 | \intertext{with e.g.\ the green triad} |
---|
| 802 | {\:}_i{\mathbb{R}_\mathrm{base}}_{1/2}^{-1/2}&= |
---|
| 803 | {\:}^{k_\mathrm{ML}}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{-1/2}. \notag |
---|
| 804 | \end{align} |
---|
| 805 | The vertical flux associated with each of these triads passes through the $w$-point |
---|
| 806 | $i,k_\mathrm{ML}-1/2$ lying \emph{below} the $i,k_\mathrm{ML}$ tracer point, |
---|
| 807 | so it is this depth |
---|
| 808 | \begin{equation} |
---|
| 809 | \label{eq:triad:zbase} |
---|
| 810 | {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2} |
---|
| 811 | \end{equation} |
---|
| 812 | (one gridbox deeper than the |
---|
| 813 | diagnosed ML depth $z_\mathrm{ML})$ that sets the $h$ used to taper |
---|
| 814 | the slopes in \eqref{eq:triad:rmtilde}. |
---|
| 815 | \item Finally, we calculate the adjusted triads |
---|
| 816 | ${\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,i_p}^{k_p}$ within the mixed |
---|
| 817 | layer, by multiplying the appropriate |
---|
| 818 | ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ by the ratio of |
---|
| 819 | the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_\mathrm{base}}_{\,i}$. For |
---|
| 820 | instance the green triad centred on $i,k$ |
---|
| 821 | \begin{align} |
---|
| 822 | {\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,1/2}^{-1/2} &= |
---|
| 823 | \frac{{z_w}_{k-1/2}}{{z_\mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{-1/2} |
---|
| 824 | \notag \\ |
---|
| 825 | \intertext{and more generally} |
---|
| 826 | {\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,i_p}^{k_p} &= |
---|
| 827 | \frac{{z_w}_{k+k_p}}{{z_\mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}.\label{eq:triad:RML} |
---|
| 828 | \end{align} |
---|
| 829 | \end{enumerate} |
---|
| 830 | |
---|
| 831 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 832 | \begin{figure}[h] |
---|
| 833 | \fcapside {\caption{\label{fig:triad:MLB_triad} Definition of |
---|
| 834 | mixed-layer depth and calculation of linearly tapered |
---|
| 835 | triads. The figure shows a water column at a given $i,j$ |
---|
| 836 | (simplified to $i$), with the ocean surface at the top. Tracer points are denoted by |
---|
| 837 | bullets, and black lines the edges of the tracer cells; $k$ |
---|
| 838 | increases upwards. \\ |
---|
| 839 | \hspace{5 em}We define the mixed-layer by setting the vertical index |
---|
| 840 | of the tracer point immediately below the mixed layer, |
---|
| 841 | $k_\mathrm{ML}$, as the maximum $k$ (shallowest tracer point) |
---|
| 842 | such that ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, |
---|
| 843 | where $i,k_{10}$ is the tracer gridbox within which the depth |
---|
| 844 | reaches 10~m. We calculate the triad slopes within the mixed |
---|
| 845 | layer by linearly tapering them from zero (at the surface) to |
---|
| 846 | the `basal' slopes, the slopes of the four triads passing through the |
---|
| 847 | $w$-point $i,k_\mathrm{ML}-1/2$ (blue square), |
---|
| 848 | ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$. Triads with |
---|
| 849 | different $i_p,k_p$, denoted by different colours, (e.g. the green |
---|
| 850 | triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.}} |
---|
[3297] | 851 | {\includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_GRIFF_MLB_triads}} |
---|
[3294] | 852 | \end{figure} |
---|
| 853 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 854 | |
---|
| 855 | \subsubsection{Additional truncation of skew iso-neutral flux |
---|
| 856 | components} |
---|
| 857 | \label{sec:triad:Gerdes-taper} |
---|
| 858 | The alternative option is activated by setting \np{ln\_triad\_iso} = |
---|
| 859 | true. This retains the same tapered slope $\rML$ described above for the |
---|
| 860 | calculation of the $_{33}$ term of the iso-neutral diffusion tensor (the |
---|
| 861 | vertical tracer flux driven by vertical tracer gradients), but |
---|
| 862 | replaces the $\rML$ in the skew term by |
---|
| 863 | \begin{equation} |
---|
| 864 | \label{eq:triad:rm*} |
---|
| 865 | \rML^*=\left.\rMLt^2\right/\tilde{r}_i-\sigma_i, |
---|
| 866 | \end{equation} |
---|
| 867 | giving a ML diffusive operator |
---|
| 868 | \begin{equation} \label{eq:triad:iso_tensor_ML2} |
---|
| 869 | D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad |
---|
| 870 | \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} |
---|
| 871 | 1 \hfill & 0 \hfill & {-\rML[1]^*}\hfill \\ |
---|
| 872 | 0 \hfill & 1 \hfill & {-\rML[2]^*} \hfill \\ |
---|
| 873 | {-\rML[1]^*}\hfill & {-\rML[2]^*} \hfill & {\rML[1]^2+\rML[2]^2} \hfill \\ |
---|
| 874 | \end{array} }} \right). |
---|
| 875 | \end{equation} |
---|
| 876 | This operator |
---|
| 877 | \footnote{To ensure good behaviour where horizontal density |
---|
| 878 | gradients are weak, we in fact follow \citet{Gerdes1991} and set |
---|
| 879 | $\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$.} |
---|
| 880 | then has the property it gives no vertical density flux, and so does |
---|
| 881 | not change the potential energy. |
---|
| 882 | This approach is similar to multiplying the iso-neutral diffusion |
---|
| 883 | coefficient by $\tilde{r}_\mathrm{max}^{-2}\tilde{r}_i^{-2}$ for steep |
---|
| 884 | slopes, as suggested by \citet{Gerdes1991} (see also \citet{Griffies_Bk04}). |
---|
| 885 | Again it is applied separately to each triad $_i^k\mathbb{R}_{i_p}^{k_p}$ |
---|
| 886 | |
---|
| 887 | In practice, this approach gives weak vertical tracer fluxes through |
---|
| 888 | the mixed-layer, as well as vanishing density fluxes. While it is |
---|
| 889 | theoretically advantageous that it does not change the potential |
---|
| 890 | energy, it may give a discontinuity between the |
---|
| 891 | fluxes within the mixed-layer (purely horizontal) and just below (along |
---|
| 892 | iso-neutral surfaces). |
---|
| 893 | % This may give strange looking results, |
---|
| 894 | % particularly where the mixed-layer depth varies strongly laterally. |
---|
[2282] | 895 | % ================================================================ |
---|
| 896 | % Skew flux formulation for Eddy Induced Velocity : |
---|
| 897 | % ================================================================ |
---|
[3294] | 898 | \section{Eddy induced advection formulated as a skew flux}\label{sec:triad:skew-flux} |
---|
[2282] | 899 | |
---|
[3294] | 900 | \subsection{The continuous skew flux formulation}\label{sec:triad:continuous-skew-flux} |
---|
| 901 | |
---|
| 902 | When Gent and McWilliams's [1990] diffusion is used, |
---|
| 903 | an additional advection term is added. The associated velocity is the so called |
---|
[2282] | 904 | eddy induced velocity, the formulation of which depends on the slopes of iso- |
---|
[3294] | 905 | neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used |
---|
| 906 | here are referenced to the geopotential surfaces, $i.e.$ \eqref{Eq_ldfslp_geo} |
---|
[2282] | 907 | is used in $z$-coordinate, and the sum \eqref{Eq_ldfslp_geo} |
---|
[3294] | 908 | + \eqref{Eq_ldfslp_iso} in $z^*$ or $s$-coordinates. |
---|
[2282] | 909 | |
---|
[3294] | 910 | The eddy induced velocity is given by: |
---|
| 911 | \begin{subequations} \label{eq:triad:eiv} |
---|
| 912 | \begin{equation}\label{eq:triad:eiv_v} |
---|
[2282] | 913 | \begin{split} |
---|
[3294] | 914 | u^* & = - \frac{1}{e_{3}}\; \partial_i\psi_1, \\ |
---|
| 915 | v^* & = - \frac{1}{e_{3}}\; \partial_j\psi_2, \\ |
---|
| 916 | w^* & = \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i \left( e_{2} \, \psi_1\right) |
---|
| 917 | + \partial_j \left( e_{1} \, \psi_2\right) \right\}, |
---|
[2282] | 918 | \end{split} |
---|
| 919 | \end{equation} |
---|
[3294] | 920 | where the streamfunctions $\psi_i$ are given by |
---|
| 921 | \begin{equation} \label{eq:triad:eiv_psi} |
---|
| 922 | \begin{split} |
---|
| 923 | \psi_1 & = A_{e} \; \tilde{r}_1, \\ |
---|
| 924 | \psi_2 & = A_{e} \; \tilde{r}_2, |
---|
| 925 | \end{split} |
---|
| 926 | \end{equation} |
---|
| 927 | \end{subequations} |
---|
| 928 | with $A_{e}$ the eddy induced velocity coefficient, and $\tilde{r}_1$ and $\tilde{r}_2$ the slopes between the iso-neutral and the geopotential surfaces. |
---|
[2282] | 929 | |
---|
[3294] | 930 | The traditional way to implement this additional advection is to add |
---|
| 931 | it to the Eulerian velocity prior to computing the tracer |
---|
| 932 | advection. This is implemented if \key{traldf\_eiv} is set in the |
---|
| 933 | default implementation, where \np{ln\_traldf\_grif} is set |
---|
| 934 | false. This allows us to take advantage of all the advection schemes |
---|
| 935 | offered for the tracers (see \S\ref{TRA_adv}) and not just a $2^{nd}$ |
---|
| 936 | order advection scheme. This is particularly useful for passive |
---|
| 937 | tracers where \emph{positivity} of the advection scheme is of |
---|
| 938 | paramount importance. |
---|
[2282] | 939 | |
---|
[3294] | 940 | However, when \np{ln\_traldf\_grif} is set true, \NEMO instead |
---|
| 941 | implements eddy induced advection according to the so-called skew form |
---|
| 942 | \citep{Griffies_JPO98}. It is based on a transformation of the advective fluxes |
---|
| 943 | using the non-divergent nature of the eddy induced velocity. |
---|
| 944 | For example in the (\textbf{i},\textbf{k}) plane, the tracer advective |
---|
| 945 | fluxes per unit area in $ijk$ space can be |
---|
[2282] | 946 | transformed as follows: |
---|
| 947 | \begin{flalign*} |
---|
| 948 | \begin{split} |
---|
[3294] | 949 | \textbf{F}_\mathrm{eiv}^T = |
---|
| 950 | \begin{pmatrix} |
---|
[2282] | 951 | {e_{2}\,e_{3}\; u^*} \\ |
---|
| 952 | {e_{1}\,e_{2}\; w^*} \\ |
---|
| 953 | \end{pmatrix} \; T |
---|
| 954 | &= |
---|
[3294] | 955 | \begin{pmatrix} |
---|
| 956 | { - \partial_k \left( e_{2} \,\psi_1 \right) \; T \;} \\ |
---|
| 957 | {+ \partial_i \left( e_{2} \, \psi_1 \right) \; T \;} \\ |
---|
[2282] | 958 | \end{pmatrix} \\ |
---|
[3294] | 959 | &= |
---|
| 960 | \begin{pmatrix} |
---|
| 961 | { - \partial_k \left( e_{2} \, \psi_1 \; T \right) \;} \\ |
---|
| 962 | {+ \partial_i \left( e_{2} \,\psi_1 \; T \right) \;} \\ |
---|
| 963 | \end{pmatrix} |
---|
| 964 | + |
---|
| 965 | \begin{pmatrix} |
---|
| 966 | {+ e_{2} \, \psi_1 \; \partial_k T} \\ |
---|
| 967 | { - e_{2} \, \psi_1 \; \partial_i T} \\ |
---|
| 968 | \end{pmatrix} |
---|
[2282] | 969 | \end{split} |
---|
| 970 | \end{flalign*} |
---|
[3294] | 971 | and since the eddy induced velocity field is non-divergent, we end up with the skew |
---|
| 972 | form of the eddy induced advective fluxes per unit area in $ijk$ space: |
---|
| 973 | \begin{equation} \label{eq:triad:eiv_skew_ijk} |
---|
| 974 | \textbf{F}_\mathrm{eiv}^T = \begin{pmatrix} |
---|
| 975 | {+ e_{2} \, \psi_1 \; \partial_k T} \\ |
---|
| 976 | { - e_{2} \, \psi_1 \; \partial_i T} \\ |
---|
[2282] | 977 | \end{pmatrix} |
---|
| 978 | \end{equation} |
---|
[3294] | 979 | The total fluxes per unit physical area are then |
---|
| 980 | \begin{equation}\label{eq:triad:eiv_skew_physical} |
---|
| 981 | \begin{split} |
---|
| 982 | f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T \\ |
---|
| 983 | f^*_2 & = \frac{1}{e_{3}}\; \psi_2 \partial_k T \\ |
---|
| 984 | f^*_3 & = -\frac{1}{e_{1}e_{2}}\; \left\{ e_{2} \psi_1 \partial_i T |
---|
| 985 | + e_{1} \psi_2 \partial_j T \right\}. \\ |
---|
| 986 | \end{split} |
---|
| 987 | \end{equation} |
---|
| 988 | Note that Eq.~ \eqref{eq:triad:eiv_skew_physical} takes the same form whatever the |
---|
| 989 | vertical coordinate, though of course the slopes |
---|
| 990 | $\tilde{r}_i$ which define the $\psi_i$ in \eqref{eq:triad:eiv_psi} are relative to geopotentials. |
---|
| 991 | The tendency associated with eddy induced velocity is then simply the convergence |
---|
| 992 | of the fluxes (\ref{eq:triad:eiv_skew_ijk}, \ref{eq:triad:eiv_skew_physical}), so |
---|
| 993 | \begin{equation} \label{eq:triad:skew_eiv_conv} |
---|
| 994 | \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 } \left[ |
---|
| 995 | \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right) |
---|
| 996 | + \frac{\partial}{\partial j} \left( e_1 \; |
---|
| 997 | \psi_2 \partial_k T\right) |
---|
| 998 | - \frac{\partial}{\partial k} \left( e_{2} \psi_1 \partial_i T |
---|
| 999 | + e_{1} \psi_2 \partial_j T \right) \right] |
---|
| 1000 | \end{equation} |
---|
| 1001 | It naturally conserves the tracer content, as it is expressed in flux |
---|
| 1002 | form. Since it has the same divergence as the advective form it also |
---|
| 1003 | preserves the tracer variance. |
---|
[2282] | 1004 | |
---|
[3294] | 1005 | \subsection{The discrete skew flux formulation} |
---|
| 1006 | The skew fluxes in (\ref{eq:triad:eiv_skew_physical}, \ref{eq:triad:eiv_skew_ijk}), like the off-diagonal terms |
---|
| 1007 | (\ref{eq:triad:i13c}, \ref{eq:triad:i31c}) of the small angle diffusion tensor, are best |
---|
| 1008 | expressed in terms of the triad slopes, as in Fig.~\ref{fig:triad:ISO_triad} |
---|
| 1009 | and Eqs~(\ref{eq:triad:i13}, \ref{eq:triad:i31}); but now in terms of the triad slopes |
---|
| 1010 | $\tilde{\mathbb{R}}$ relative to geopotentials instead of the |
---|
| 1011 | $\mathbb{R}$ relative to coordinate surfaces. The discrete form of |
---|
| 1012 | \eqref{eq:triad:eiv_skew_ijk} using the slopes \eqref{eq:triad:R} and |
---|
| 1013 | defining $A_e$ at $T$-points is then given by: |
---|
[2282] | 1014 | |
---|
| 1015 | |
---|
[3294] | 1016 | \begin{subequations}\label{eq:triad:allskewflux} |
---|
| 1017 | \begin{flalign}\label{eq:triad:vect_skew_flux} |
---|
| 1018 | \vect{F}_\mathrm{eiv}(T) &\equiv |
---|
| 1019 | \sum_{\substack{i_p,\,k_p}} |
---|
| 1020 | \begin{pmatrix} |
---|
| 1021 | {_{i+1/2-i_p}^k {\mathbb{S}_u}_{i_p}^{k_p} } (T) \\ |
---|
| 1022 | \\ |
---|
| 1023 | {_i^{k+1/2-k_p} {\mathbb{S}_w}_{i_p}^{k_p} } (T) \\ |
---|
| 1024 | \end{pmatrix}, |
---|
| 1025 | \end{flalign} |
---|
| 1026 | where the skew flux in the $i$-direction associated with a given |
---|
| 1027 | triad is (\ref{eq:triad:latflux-triad}, \ref{eq:triad:triadfluxu}): |
---|
| 1028 | \begin{align} |
---|
| 1029 | \label{eq:triad:skewfluxu} |
---|
| 1030 | _i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) &= + \quarter {A_e}_i^k{ |
---|
| 1031 | \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
| 1032 | \ {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} \ |
---|
| 1033 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }, |
---|
| 1034 | \\ |
---|
| 1035 | \intertext{and \eqref{eq:triad:triadfluxw} in the $k$-direction, changing the sign |
---|
| 1036 | to be consistent with \eqref{eq:triad:eiv_skew_ijk}:} |
---|
| 1037 | _i^k {\mathbb{S}_w}_{i_p}^{k_p} (T) |
---|
| 1038 | &= -\quarter {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}} |
---|
| 1039 | {_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} |
---|
| 1040 | \end{align} |
---|
| 1041 | \end{subequations} |
---|
[2282] | 1042 | |
---|
[3294] | 1043 | Such a discretisation is consistent with the iso-neutral |
---|
| 1044 | operator as it uses the same definition for the slopes. It also |
---|
| 1045 | ensures the following two key properties. |
---|
| 1046 | \subsubsection{No change in tracer variance} |
---|
| 1047 | The discretization conserves tracer variance, $i.e.$ it does not |
---|
| 1048 | include a diffusive component but is a `pure' advection term. This can |
---|
| 1049 | be seen |
---|
| 1050 | %either from Appendix \ref{Apdx_eiv_skew} or |
---|
| 1051 | by considering the |
---|
| 1052 | fluxes associated with a given triad slope |
---|
| 1053 | $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. For, following |
---|
| 1054 | \S\ref{sec:triad:variance} and \eqref{eq:triad:dvar_iso_i}, the |
---|
| 1055 | associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$ |
---|
| 1056 | drives a net rate of change of variance, summed over the two |
---|
| 1057 | $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of |
---|
| 1058 | \begin{equation} |
---|
| 1059 | \label{eq:triad:dvar_eiv_i} |
---|
| 1060 | _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], |
---|
| 1061 | \end{equation} |
---|
| 1062 | while the associated vertical skew-flux gives a variance change summed over the |
---|
| 1063 | $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of |
---|
| 1064 | \begin{equation} |
---|
| 1065 | \label{eq:triad:dvar_eiv_k} |
---|
| 1066 | _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. |
---|
| 1067 | \end{equation} |
---|
| 1068 | Inspection of the definitions (\ref{eq:triad:skewfluxu}, \ref{eq:triad:skewfluxw}) |
---|
| 1069 | shows that these two variance changes (\ref{eq:triad:dvar_eiv_i}, \ref{eq:triad:dvar_eiv_k}) |
---|
| 1070 | sum to zero. Hence the two fluxes associated with each triad make no |
---|
| 1071 | net contribution to the variance budget. |
---|
[2282] | 1072 | |
---|
[3294] | 1073 | \subsubsection{Reduction in gravitational PE} |
---|
| 1074 | The vertical density flux associated with the vertical skew-flux |
---|
| 1075 | always has the same sign as the vertical density gradient; thus, so |
---|
| 1076 | long as the fluid is stable (the vertical density gradient is |
---|
| 1077 | negative) the vertical density flux is negative (downward) and hence |
---|
| 1078 | reduces the gravitational PE. |
---|
[2282] | 1079 | |
---|
[3294] | 1080 | For the change in gravitational PE driven by the $k$-flux is |
---|
| 1081 | \begin{align} |
---|
| 1082 | \label{eq:triad:vert_densityPE} |
---|
| 1083 | g {e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) |
---|
| 1084 | &=g {e_{3w}}_{\,i}^{\,k+k_p}\left[-\alpha _i^k {\:}_i^k |
---|
| 1085 | {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k |
---|
| 1086 | {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\ |
---|
| 1087 | \intertext{Substituting ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from |
---|
| 1088 | \eqref{eq:triad:skewfluxw}, gives} |
---|
| 1089 | % and separating out |
---|
| 1090 | % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$, |
---|
| 1091 | % gives two terms. The |
---|
| 1092 | % first $\rtriad{R}$ term (the only term for $z$-coordinates) is: |
---|
| 1093 | &=-\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} |
---|
| 1094 | \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 \\ |
---|
| 1095 | &=+\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
| 1096 | \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}} |
---|
| 1097 | \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}}, |
---|
| 1098 | \end{align} |
---|
| 1099 | using the definition of the triad slope $\rtriad{R}$, |
---|
| 1100 | \eqref{eq:triad:R} to express $-\alpha _i^k\delta_{i+ i_p}[T^k]+ |
---|
| 1101 | \beta_i^k\delta_{i+ i_p}[S^k]$ in terms of $-\alpha_i^k \delta_{k+ |
---|
| 1102 | k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$. |
---|
[2282] | 1103 | |
---|
[3294] | 1104 | Where the coordinates slope, the $i$-flux gives a PE change |
---|
| 1105 | \begin{multline} |
---|
| 1106 | \label{eq:triad:lat_densityPE} |
---|
| 1107 | g \delta_{i+i_p}[z_T^k] |
---|
| 1108 | \left[ |
---|
| 1109 | -\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) |
---|
| 1110 | \right] \\ |
---|
| 1111 | = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
| 1112 | \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
| 1113 | \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) |
---|
| 1114 | \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}}, |
---|
| 1115 | \end{multline} |
---|
| 1116 | (using \eqref{eq:triad:skewfluxu}) and so the total PE change |
---|
| 1117 | \eqref{eq:triad:vert_densityPE} + \eqref{eq:triad:lat_densityPE} associated with the triad fluxes is |
---|
| 1118 | \begin{multline} |
---|
| 1119 | \label{eq:triad:tot_densityPE} |
---|
| 1120 | g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) + |
---|
| 1121 | g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\ |
---|
| 1122 | = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
| 1123 | \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 |
---|
| 1124 | \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}}. |
---|
| 1125 | \end{multline} |
---|
| 1126 | Where the fluid is stable, with $-\alpha_i^k \delta_{k+ k_p}[T^i]+ |
---|
| 1127 | \beta_i^k\delta_{k+ k_p}[S^i]<0$, this PE change is negative. |
---|
[2282] | 1128 | |
---|
[3294] | 1129 | \subsection{Treatment of the triads at the boundaries}\label{sec:triad:skew_bdry} |
---|
| 1130 | Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes |
---|
| 1131 | are masked at the boundaries in exactly the same way as are the triad |
---|
| 1132 | slopes \rtriad{R} used for the iso-neutral diffusive fluxes, as |
---|
| 1133 | described in \S\ref{sec:triad:iso_bdry} and |
---|
| 1134 | Fig.~\ref{fig:triad:bdry_triads}. Thus surface layer triads |
---|
| 1135 | $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are |
---|
| 1136 | masked, and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ |
---|
| 1137 | and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the |
---|
| 1138 | $i,k+1$ or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ |
---|
| 1139 | $u$-point is masked. The namelist parameter \np{ln\_botmix\_grif} has |
---|
| 1140 | no effect on the eddy-induced skew-fluxes. |
---|
[2282] | 1141 | |
---|
[3294] | 1142 | \subsection{ Limiting of the slopes within the interior}\label{sec:triad:limitskew} |
---|
| 1143 | Presently, the iso-neutral slopes $\tilde{r}_i$ relative |
---|
| 1144 | to geopotentials are limited to be less than $1/100$, exactly as in |
---|
| 1145 | calculating the iso-neutral diffusion, \S \ref{sec:triad:limit}. Each |
---|
| 1146 | individual triad \rtriadt{R} is so limited. |
---|
[2282] | 1147 | |
---|
[3294] | 1148 | \subsection{Tapering within the surface mixed layer}\label{sec:triad:taperskew} |
---|
| 1149 | The slopes $\tilde{r}_i$ relative to |
---|
| 1150 | geopotentials (and thus the individual triads \rtriadt{R}) are always tapered linearly from their value immediately below the mixed layer to zero at the |
---|
| 1151 | surface \eqref{eq:triad:rmtilde}, as described in \S\ref{sec:triad:lintaper}. This is |
---|
| 1152 | option (c) of Fig.~\ref{Fig_eiv_slp}. This linear tapering for the |
---|
| 1153 | slopes used to calculate the eddy-induced fluxes is |
---|
| 1154 | unaffected by the value of \np{ln\_triad\_iso}. |
---|
[2282] | 1155 | |
---|
[3294] | 1156 | The justification for this linear slope tapering is that, for $A_e$ |
---|
| 1157 | that is constant or varies only in the horizontal (the most commonly |
---|
| 1158 | used options in \NEMO: see \S\ref{LDF_coef}), it is |
---|
| 1159 | equivalent to a horizontal eiv (eddy-induced velocity) that is uniform |
---|
| 1160 | within the mixed layer \eqref{eq:triad:eiv_v}. This ensures that the |
---|
| 1161 | eiv velocities do not restratify the mixed layer \citep{Treguier1997, |
---|
| 1162 | Danabasoglu_al_2008}. Equivantly, in terms |
---|
| 1163 | of the skew-flux formulation we use here, the |
---|
| 1164 | linear slope tapering within the mixed-layer gives a linearly varying |
---|
| 1165 | vertical flux, and so a tracer convergence uniform in depth (the |
---|
| 1166 | horizontal flux convergence is relatively insignificant within the mixed-layer). |
---|
| 1167 | |
---|
| 1168 | \subsection{Streamfunction diagnostics}\label{sec:triad:sfdiag} |
---|
| 1169 | Where the namelist parameter \np{ln\_traldf\_gdia}=true, diagnosed |
---|
| 1170 | mean eddy-induced velocities are output. Each time step, |
---|
| 1171 | streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at |
---|
| 1172 | $uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$ |
---|
| 1173 | (integer $i$, integer +1/2 $j$, integer +1/2 $k$) points (see Table |
---|
| 1174 | \ref{Tab_cell}) respectively. We follow \citep{Griffies_Bk04} and |
---|
| 1175 | calculate the streamfunction at a given $uw$-point from the |
---|
| 1176 | surrounding four triads according to: |
---|
| 1177 | \begin{equation} |
---|
| 1178 | \label{eq:triad:sfdiagi} |
---|
| 1179 | {\psi_1}_{i+1/2}^{k+1/2}={\quarter}\sum_{\substack{i_p,\,k_p}} |
---|
| 1180 | {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}. |
---|
| 1181 | \end{equation} |
---|
| 1182 | The streamfunction $\psi_1$ is calculated similarly at $vw$ points. |
---|
| 1183 | The eddy-induced velocities are then calculated from the |
---|
| 1184 | straightforward discretisation of \eqref{eq:triad:eiv_v}: |
---|
| 1185 | \begin{equation}\label{eq:triad:eiv_v_discrete} |
---|
| 1186 | \begin{split} |
---|
| 1187 | {u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k}}\left({\psi_1}_{i+1/2}^{k+1/2}-{\psi_1}_{i+1/2}^{k+1/2}\right), \\ |
---|
| 1188 | {v^*}_{j+1/2}^{k} & = - \frac{1}{{e_{3v}}_{j}^{k}}\left({\psi_2}_{j+1/2}^{k+1/2}-{\psi_2}_{j+1/2}^{k+1/2}\right), \\ |
---|
| 1189 | {w^*}_{i,j}^{k+1/2} & = \frac{1}{e_{1t}e_{2t}}\; \left\{ |
---|
| 1190 | {e_{2u}}_{i+1/2}^{k+1/2} \,{\psi_1}_{i+1/2}^{k+1/2} - |
---|
| 1191 | {e_{2u}}_{i-1/2}^{k+1/2} \,{\psi_1}_{i-1/2}^{k+1/2} \right. + \\ |
---|
| 1192 | \phantom{=} & \qquad\qquad\left. {e_{2v}}_{j+1/2}^{k+1/2} \,{\psi_2}_{j+1/2}^{k+1/2} - {e_{2v}}_{j-1/2}^{k+1/2} \,{\psi_2}_{j-1/2}^{k+1/2} \right\}, |
---|
| 1193 | \end{split} |
---|
| 1194 | \end{equation} |
---|