[2282] | 1 | % ================================================================ |
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[3294] | 2 | % Iso-neutral diffusion : |
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[2282] | 3 | % ================================================================ |
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[3294] | 4 | \chapter[Iso-neutral diffusion and eddy advection using |
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| 5 | triads]{Iso-neutral diffusion and eddy advection using triads} |
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| 6 | \label{sec:triad} |
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[2282] | 7 | \minitoc |
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[3294] | 8 | \pagebreak |
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[4147] | 9 | \section{Choice of \ngn{namtra\_ldf} namelist parameters} |
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[3294] | 10 | %-----------------------------------------nam_traldf------------------------------------------------------ |
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| 11 | \namdisplay{namtra_ldf} |
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| 12 | %--------------------------------------------------------------------------------------------------------- |
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[2282] | 13 | |
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[6289] | 14 | Two scheme are available to perform the iso-neutral diffusion. |
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| 15 | If the namelist logical \np{ln\_traldf\_triad} is set true, |
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| 16 | \NEMO updates both active and passive tracers using the Griffies triad representation |
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| 17 | of iso-neutral diffusion and the eddy-induced advective skew (GM) fluxes. |
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| 18 | If the namelist logical \np{ln\_traldf\_iso} is set true, |
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| 19 | the filtered version of Cox's original scheme (the Standard scheme) is employed (\S\ref{LDF_slp}). |
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| 20 | In the present implementation of the Griffies scheme, |
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| 21 | the advective skew fluxes are implemented even if \np{ln\_traldf\_eiv} is false. |
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| 22 | |
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[3294] | 23 | Values of iso-neutral diffusivity and GM coefficient are set as |
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[6289] | 24 | described in \S\ref{LDF_coef}. Note that when GM fluxes are used, |
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| 25 | the eddy-advective (GM) velocities are output for diagnostic purposes using xIOS, |
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| 26 | even though the eddy advection is accomplished by means of the skew fluxes. |
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[2282] | 27 | |
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[6289] | 28 | |
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[3294] | 29 | The options specific to the Griffies scheme include: |
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| 30 | \begin{description}[font=\normalfont] |
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[6289] | 31 | \item[\np{ln\_triad\_iso}] See \S\ref{sec:triad:taper}. If this is set false (the default), then |
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[3294] | 32 | `iso-neutral' mixing is accomplished within the surface mixed-layer |
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| 33 | along slopes linearly decreasing with depth from the value immediately below |
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[6289] | 34 | the mixed-layer to zero (flat) at the surface (\S\ref{sec:triad:lintaper}). |
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| 35 | This is the same treatment as used in the default implementation \S\ref{LDF_slp_iso}; Fig.~\ref{Fig_eiv_slp}. |
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| 36 | Where \np{ln\_triad\_iso} is set true, the vertical skew flux is further reduced |
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| 37 | to ensure no vertical buoyancy flux, giving an almost pure |
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[3294] | 38 | horizontal diffusive tracer flux within the mixed layer. This is similar to |
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| 39 | the tapering suggested by \citet{Gerdes1991}. See \S\ref{sec:triad:Gerdes-taper} |
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[6289] | 40 | \item[\np{ln\_botmix\_triad}] See \S\ref{sec:triad:iso_bdry}. |
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| 41 | If this is set false (the default) then the lateral diffusive fluxes |
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| 42 | associated with triads partly masked by topography are neglected. |
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| 43 | If it is set true, however, then these lateral diffusive fluxes are applied, |
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| 44 | giving smoother bottom tracer fields at the cost of introducing diapycnal mixing. |
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| 45 | \item[\np{rn\_sw\_triad}] blah blah to be added.... |
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[3294] | 46 | \end{description} |
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[6289] | 47 | The options shared with the Standard scheme include: |
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| 48 | \begin{description}[font=\normalfont] |
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| 49 | \item[\np{ln\_traldf\_msc}] blah blah to be added |
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| 50 | \item[\np{rn\_slpmax}] blah blah to be added |
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| 51 | \end{description} |
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[3294] | 52 | \section{Triad formulation of iso-neutral diffusion} |
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| 53 | \label{sec:triad:iso} |
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[6289] | 54 | We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98}, |
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| 55 | but formulated within the \NEMO framework, using scale factors rather than grid-sizes. |
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[2282] | 56 | |
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[3294] | 57 | \subsection{The iso-neutral diffusion operator} |
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| 58 | The iso-neutral second order tracer diffusive operator for small |
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| 59 | angles between iso-neutral surfaces and geopotentials is given by |
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| 60 | \eqref{Eq_PE_iso_tensor}: |
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| 61 | \begin{subequations} \label{eq:triad:PE_iso_tensor} |
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| 62 | \begin{equation} |
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| 63 | D^{lT}=-\Div\vect{f}^{lT}\equiv |
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| 64 | -\frac{1}{e_1e_2e_3}\left[\pd{i}\left (f_1^{lT}e_2e_3\right) + |
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| 65 | \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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| 66 | \end{equation} |
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| 67 | where the diffusive flux per unit area of physical space |
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| 68 | \begin{equation} |
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| 69 | \vect{f}^{lT}=-\Alt\Re\cdot\grad T, |
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| 70 | \end{equation} |
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| 71 | \begin{equation} |
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| 72 | \label{eq:triad:PE_iso_tensor:c} |
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| 73 | \mbox{with}\quad \;\;\Re = |
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| 74 | \begin{pmatrix} |
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[6289] | 75 | 1 & 0 & -r_1 \mystrut \\ |
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| 76 | 0 & 1 & -r_2 \mystrut \\ |
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| 77 | -r_1 & -r_2 & r_1 ^2+r_2 ^2 \mystrut |
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[3294] | 78 | \end{pmatrix} |
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| 79 | \quad \text{and} \quad\grad T= |
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| 80 | \begin{pmatrix} |
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[6289] | 81 | \frac{1}{e_1} \pd[T]{i} \mystrut \\ |
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| 82 | \frac{1}{e_2} \pd[T]{j} \mystrut \\ |
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| 83 | \frac{1}{e_3} \pd[T]{k} \mystrut |
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[3294] | 84 | \end{pmatrix}. |
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| 85 | \end{equation} |
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| 86 | \end{subequations} |
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| 87 | % \left( {{\begin{array}{*{20}c} |
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| 88 | % 1 \hfill & 0 \hfill & {-r_1 } \hfill \\ |
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| 89 | % 0 \hfill & 1 \hfill & {-r_2 } \hfill \\ |
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| 90 | % {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ |
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| 91 | % \end{array} }} \right) |
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[6289] | 92 | Here \eqref{Eq_PE_iso_slopes} |
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[2282] | 93 | \begin{align*} |
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| 94 | r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i} |
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| 95 | \right) |
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| 96 | \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \\ |
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| 97 | &=-\frac{e_3 }{e_1 } \left( -\alpha\frac{\partial T }{\partial i} + |
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| 98 | \beta\frac{\partial S }{\partial i} \right) \left( |
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| 99 | -\alpha\frac{\partial T }{\partial k} + \beta\frac{\partial S |
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| 100 | }{\partial k} \right)^{-1} |
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| 101 | \end{align*} |
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[3294] | 102 | is the $i$-component of the slope of the iso-neutral surface relative to the computational |
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| 103 | surface, and $r_2$ is the $j$-component. |
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[2282] | 104 | |
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[3294] | 105 | We will find it useful to consider the fluxes per unit area in $i,j,k$ |
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| 106 | space; we write |
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| 107 | \begin{equation} |
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| 108 | \label{eq:triad:Fijk} |
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| 109 | \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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| 110 | \end{equation} |
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| 111 | Additionally, we will sometimes write the contributions towards the |
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| 112 | fluxes $\vect{f}$ and $\vect{F}_\mathrm{iso}$ from the component |
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| 113 | $R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$, with |
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| 114 | $f_{ij}=R_{ij}e_i^{-1}\partial T/\partial x_i$ (no summation) etc. |
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| 115 | |
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| 116 | The off-diagonal terms of the small angle diffusion tensor |
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| 117 | \eqref{Eq_PE_iso_tensor}, \eqref{eq:triad:PE_iso_tensor:c} produce skew-fluxes along the |
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| 118 | $i$- and $j$-directions resulting from the vertical tracer gradient: |
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| 119 | \begin{align} |
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| 120 | \label{eq:triad:i13c} |
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| 121 | 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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| 122 | \intertext{and in the k-direction resulting from the lateral tracer gradients} |
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| 123 | \label{eq:triad:i31c} |
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| 124 | 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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| 125 | \end{align} |
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| 126 | |
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| 127 | The vertical diffusive flux associated with the $_{33}$ |
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[2282] | 128 | component of the small angle diffusion tensor is |
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| 129 | \begin{equation} |
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[3294] | 130 | \label{eq:triad:i33c} |
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| 131 | f_{33}=-\Alt(r_1^2 +r_2^2) \frac{1}{e_3}\frac{\partial T}{\partial k}. |
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[2282] | 132 | \end{equation} |
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| 133 | |
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| 134 | Since there are no cross terms involving $r_1$ and $r_2$ in the above, we can |
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[3294] | 135 | consider the iso-neutral diffusive fluxes separately in the $i$-$k$ and $j$-$k$ |
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[2282] | 136 | planes, just adding together the vertical components from each |
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[3294] | 137 | plane. The following description will describe the fluxes on the $i$-$k$ |
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[2282] | 138 | plane. |
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| 139 | |
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[3294] | 140 | There is no natural discretization for the $i$-component of the |
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| 141 | skew-flux, \eqref{eq:triad:i13c}, as |
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| 142 | although it must be evaluated at $u$-points, it involves vertical |
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[2282] | 143 | gradients (both for the tracer and the slope $r_1$), defined at |
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[3294] | 144 | $w$-points. Similarly, the vertical skew flux, \eqref{eq:triad:i31c}, is evaluated at |
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| 145 | $w$-points but involves horizontal gradients defined at $u$-points. |
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[2282] | 146 | |
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| 147 | \subsection{The standard discretization} |
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| 148 | The straightforward approach to discretize the lateral skew flux |
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[3294] | 149 | \eqref{eq:triad:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 |
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[2282] | 150 | into OPA, \eqref{Eq_tra_ldf_iso}, is to calculate a mean vertical |
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[3294] | 151 | gradient at the $u$-point from the average of the four surrounding |
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[2282] | 152 | vertical tracer gradients, and multiply this by a mean slope at the |
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[3294] | 153 | $u$-point, calculated from the averaged surrounding vertical density |
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| 154 | gradients. The total area-integrated skew-flux (flux per unit area in |
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| 155 | $ijk$ space) from tracer cell $i,k$ |
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| 156 | to $i+1,k$, noting that the $e_{{3}_{i+1/2}^k}$ in the area |
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| 157 | $e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with |
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| 158 | the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer |
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[2282] | 159 | gradient, is then \eqref{Eq_tra_ldf_iso} |
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| 160 | \begin{equation*} |
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[3294] | 161 | \left(F_u^{13} \right)_{i+\hhalf}^k = \Alts_{i+\hhalf}^k |
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| 162 | {e_{2}}_{i+1/2}^k \overline{\overline |
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[2282] | 163 | r_1} ^{\,i,k}\,\overline{\overline{\delta_k T}}^{\,i,k}, |
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| 164 | \end{equation*} |
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| 165 | where |
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| 166 | \begin{equation*} |
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| 167 | \overline{\overline |
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[3294] | 168 | r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k} |
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| 169 | \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}}, |
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[2282] | 170 | \end{equation*} |
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[3294] | 171 | and here and in the following we drop the $^{lT}$ superscript from |
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| 172 | $\Alt$ for simplicity. |
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[2282] | 173 | Unfortunately the resulting combination $\overline{\overline{\delta_k |
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| 174 | \bullet}}^{\,i,k}$ of a $k$ average and a $k$ difference %of the tracer |
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| 175 | reduces to $\bullet_{k+1}-\bullet_{k-1}$, so two-grid-point oscillations are |
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| 176 | invisible to this discretization of the iso-neutral operator. These |
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| 177 | \emph{computational modes} will not be damped by this operator, and |
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| 178 | may even possibly be amplified by it. Consequently, applying this |
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| 179 | operator to a tracer does not guarantee the decrease of its |
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| 180 | global-average variance. To correct this, we introduced a smoothing of |
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| 181 | the slopes of the iso-neutral surfaces (see \S\ref{LDF}). This |
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[3294] | 182 | technique works for $T$ and $S$ in so far as they are active tracers |
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[2282] | 183 | ($i.e.$ they enter the computation of density), but it does not work |
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| 184 | for a passive tracer. |
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| 185 | \subsection{Expression of the skew-flux in terms of triad slopes} |
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| 186 | \citep{Griffies_al_JPO98} introduce a different discretization of the |
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| 187 | off-diagonal terms that nicely solves the problem. |
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[3294] | 188 | % Instead of multiplying the mean slope calculated at the $u$-point by |
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| 189 | % the mean vertical gradient at the $u$-point, |
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[2282] | 190 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[6289] | 191 | \begin{figure}[tb] \begin{center} |
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[3297] | 192 | \includegraphics[width=1.05\textwidth]{./TexFiles/Figures/Fig_GRIFF_triad_fluxes} |
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[3294] | 193 | \caption{ \label{fig:triad:ISO_triad} |
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[2376] | 194 | (a) Arrangement of triads $S_i$ and tracer gradients to |
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[3294] | 195 | give lateral tracer flux from box $i,k$ to $i+1,k$ |
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[2376] | 196 | (b) Triads $S'_i$ and tracer gradients to give vertical tracer flux from |
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| 197 | box $i,k$ to $i,k+1$.} |
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[3294] | 198 | \end{center} \end{figure} |
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[2282] | 199 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 200 | They get the skew flux from the products of the vertical gradients at |
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[3294] | 201 | each $w$-point surrounding the $u$-point with the corresponding `triad' |
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| 202 | slope calculated from the lateral density gradient across the $u$-point |
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| 203 | divided by the vertical density gradient at the same $w$-point as the |
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| 204 | tracer gradient. See Fig.~\ref{fig:triad:ISO_triad}a, where the thick lines |
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[2282] | 205 | denote the tracer gradients, and the thin lines the corresponding |
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| 206 | triads, with slopes $s_1, \dotsc s_4$. The total area-integrated |
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| 207 | skew-flux from tracer cell $i,k$ to $i+1,k$ |
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| 208 | \begin{multline} |
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[3294] | 209 | \label{eq:triad:i13} |
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| 210 | \left( F_u^{13} \right)_{i+\frac{1}{2}}^k = \Alts_{i+1}^k a_1 s_1 |
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[2282] | 211 | \delta _{k+\frac{1}{2}} \left[ T^{i+1} |
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[3294] | 212 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} + \Alts _i^k a_2 s_2 \delta |
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[2282] | 213 | _{k+\frac{1}{2}} \left[ T^i |
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| 214 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} \\ |
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[3294] | 215 | +\Alts _{i+1}^k a_3 s_3 \delta _{k-\frac{1}{2}} \left[ T^{i+1} |
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| 216 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} +\Alts _i^k a_4 s_4 \delta |
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[2282] | 217 | _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}, |
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| 218 | \end{multline} |
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| 219 | where the contributions of the triad fluxes are weighted by areas |
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[3294] | 220 | $a_1, \dotsc a_4$, and $\Alts$ is now defined at the tracer points |
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| 221 | rather than the $u$-points. This discretization gives a much closer |
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[2282] | 222 | stencil, and disallows the two-point computational modes. |
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| 223 | |
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[3294] | 224 | The vertical skew flux \eqref{eq:triad:i31c} from tracer cell $i,k$ to $i,k+1$ at the |
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| 225 | $w$-point $i,k+\hhalf$ is constructed similarly (Fig.~\ref{fig:triad:ISO_triad}b) |
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[2282] | 226 | by multiplying lateral tracer gradients from each of the four |
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[3294] | 227 | surrounding $u$-points by the appropriate triad slope: |
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[2282] | 228 | \begin{multline} |
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[3294] | 229 | \label{eq:triad:i31} |
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| 230 | \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} = \Alts_i^{k+1} a_{1}' |
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[2282] | 231 | s_{1}' \delta _{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1} |
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[3294] | 232 | +\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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| 233 | + \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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| 234 | +\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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[2282] | 235 | \end{multline} |
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[3294] | 236 | |
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| 237 | We notate the triad slopes $s_i$ and $s'_i$ in terms of the `anchor point' $i,k$ |
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| 238 | (appearing in both the vertical and lateral gradient), and the $u$- and |
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| 239 | $w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the |
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| 240 | triad as follows (see also Fig.~\ref{fig:triad:ISO_triad}): |
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[2282] | 241 | \begin{equation} |
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[3294] | 242 | \label{eq:triad:R} |
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| 243 | _i^k \mathbb{R}_{i_p}^{k_p} |
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| 244 | =-\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} |
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[2282] | 245 | \ |
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[3294] | 246 | \frac |
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[6289] | 247 | { \alpha_i^k \ \delta_{i+i_p}[T^k] - \beta_i^k \ \delta_{i+i_p}[S^k] } |
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| 248 | { \alpha_i^k \ \delta_{k+k_p}[T^i] - \beta_i^k \ \delta_{k+k_p}[S^i] }. |
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[2282] | 249 | \end{equation} |
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[6289] | 250 | In calculating the slopes of the local neutral surfaces, |
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| 251 | the expansion coefficients $\alpha$ and $\beta$ are evaluated at the anchor points of the triad, |
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| 252 | while the metrics are calculated at the $u$- and $w$-points on the arms. |
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[2282] | 253 | |
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| 254 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[6289] | 255 | \begin{figure}[tb] \begin{center} |
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[3297] | 256 | \includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_GRIFF_qcells} |
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[3294] | 257 | \caption{ \label{fig:triad:qcells} |
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[3297] | 258 | Triad notation for quarter cells. $T$-cells are inside |
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[3294] | 259 | boxes, while the $i+\half,k$ $u$-cell is shaded in green and the |
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| 260 | $i,k+\half$ $w$-cell is shaded in pink.} |
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[2282] | 261 | \end{center} \end{figure} |
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| 262 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 263 | |
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[6289] | 264 | Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (Fig.~\ref{fig:triad:qcells}) with the quarter |
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| 265 | cell that is the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ $u$-cell and the $i,k+k_p$ $w$-cell. |
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| 266 | Expressing the slopes $s_i$ and $s'_i$ in \eqref{eq:triad:i13} and \eqref{eq:triad:i31} in this notation, |
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| 267 | we have $e.g.$ \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. |
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| 268 | Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) |
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| 269 | to calculate the lateral flux along its $u$-arm, at $(i+i_p,k)$, |
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| 270 | and then again as an $s'$ to calculate the vertical flux along its $w$-arm at $(i,k+k_p)$. |
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| 271 | Each vertical area $a_i$ used to calculate the lateral flux and horizontal area $a'_i$ used |
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| 272 | to calculate the vertical flux can also be identified as the area across the $u$- and $w$-arms |
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| 273 | of a unique triad, and we notate these areas, similarly to the triad slopes, |
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| 274 | as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, |
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| 275 | where $e.g.$ in \eqref{eq:triad:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, |
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| 276 | and in \eqref{eq:triad:i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. |
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[2282] | 277 | |
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| 278 | \subsection{The full triad fluxes} |
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[3294] | 279 | A key property of iso-neutral diffusion is that it should not affect |
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[2282] | 280 | the (locally referenced) density. In particular there should be no |
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| 281 | lateral or vertical density flux. The lateral density flux disappears so long as the |
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| 282 | area-integrated lateral diffusive flux from tracer cell $i,k$ to |
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| 283 | $i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the |
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| 284 | form |
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| 285 | \begin{equation} |
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[3294] | 286 | \label{eq:triad:i11} |
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[2282] | 287 | \left( F_u^{11} \right) _{i+\frac{1}{2}} ^{k} = |
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[3294] | 288 | - \left( \Alts_i^{k+1} a_{1} + \Alts_i^{k+1} a_{2} + \Alts_i^k |
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| 289 | a_{3} + \Alts_i^k a_{4} \right) |
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[2282] | 290 | \frac{\delta _{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}}, |
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| 291 | \end{equation} |
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[3294] | 292 | where the areas $a_i$ are as in \eqref{eq:triad:i13}. In this case, |
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| 293 | separating the total lateral flux, the sum of \eqref{eq:triad:i13} and |
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| 294 | \eqref{eq:triad:i11}, into triad components, a lateral tracer |
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[2282] | 295 | flux |
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| 296 | \begin{equation} |
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[3294] | 297 | \label{eq:triad:latflux-triad} |
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| 298 | _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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[2282] | 299 | \left( |
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| 300 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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| 301 | -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
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| 302 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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| 303 | \right) |
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| 304 | \end{equation} |
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| 305 | can be identified with each triad. Then, because the |
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| 306 | same metric factors ${e_{3w}}_{\,i}^{\,k+k_p}$ and |
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| 307 | ${e_{1u}}_{\,i+i_p}^{\,k}$ are employed for both the density gradients |
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| 308 | in $ _i^k \mathbb{R}_{i_p}^{k_p}$ and the tracer gradients, the lateral |
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| 309 | density flux associated with each triad separately disappears. |
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| 310 | \begin{equation} |
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[3294] | 311 | \label{eq:triad:latflux-rho} |
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[2282] | 312 | {\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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| 313 | \end{equation} |
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| 314 | Thus the total flux $\left( F_u^{31} \right) ^i _{i,k+\frac{1}{2}} + |
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| 315 | \left( F_u^{11} \right) ^i _{i,k+\frac{1}{2}}$ from tracer cell $i,k$ |
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| 316 | to $i+1,k$ must also vanish since it is a sum of four such triad fluxes. |
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| 317 | |
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[3294] | 318 | The squared slope $r_1^2$ in the expression \eqref{eq:triad:i33c} for the |
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[2282] | 319 | $_{33}$ component is also expressed in terms of area-weighted |
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| 320 | squared triad slopes, so the area-integrated vertical flux from tracer |
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| 321 | cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is |
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| 322 | \begin{equation} |
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[3294] | 323 | \label{eq:triad:i33} |
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[2282] | 324 | \left( F_w^{33} \right) _i^{k+\frac{1}{2}} = |
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[3294] | 325 | - \left( \Alts_i^{k+1} a_{1}' s_{1}'^2 |
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| 326 | + \Alts_i^{k+1} a_{2}' s_{2}'^2 |
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| 327 | + \Alts_i^k a_{3}' s_{3}'^2 |
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| 328 | + \Alts_i^k a_{4}' s_{4}'^2 \right)\delta_{k+\frac{1}{2}} \left[ T^{i+1} \right], |
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[2282] | 329 | \end{equation} |
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| 330 | where the areas $a'$ and slopes $s'$ are the same as in |
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[3294] | 331 | \eqref{eq:triad:i31}. |
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| 332 | Then, separating the total vertical flux, the sum of \eqref{eq:triad:i31} and |
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| 333 | \eqref{eq:triad:i33}, into triad components, a vertical flux |
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[2282] | 334 | \begin{align} |
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[3294] | 335 | \label{eq:triad:vertflux-triad} |
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[2282] | 336 | _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) |
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[3294] | 337 | &= \Alts_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p} |
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[2282] | 338 | \left( |
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| 339 | {_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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| 340 | -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ |
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| 341 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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| 342 | \right) \\ |
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| 343 | &= - \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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[3294] | 344 | {_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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[2282] | 345 | \end{align} |
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| 346 | may be associated with each triad. Each vertical density flux $_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)$ |
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| 347 | associated with a triad then separately disappears (because the |
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| 348 | lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (\rho)$ |
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| 349 | disappears). Consequently the total vertical density flux $\left( F_w^{31} \right)_i ^{k+\frac{1}{2}} + |
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| 350 | \left( F_w^{33} \right)_i^{k+\frac{1}{2}}$ from tracer cell $i,k$ |
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| 351 | to $i,k+1$ must also vanish since it is a sum of four such triad |
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| 352 | fluxes. |
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| 353 | |
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[3294] | 354 | 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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| 355 | the $u$-fluxes and $w$-fluxes in |
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| 356 | \eqref{eq:triad:i31}, \eqref{eq:triad:i13}, \eqref{eq:triad:i11} \eqref{eq:triad:i33} and |
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| 357 | Fig.~\ref{fig:triad:ISO_triad} to write out the iso-neutral fluxes at $u$- and |
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[2282] | 358 | $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces: |
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| 359 | %(Fig.~\ref{Fig_ISO_triad}): |
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[3294] | 360 | \begin{flalign} \label{Eq_iso_flux} \vect{F}_\mathrm{iso}(T) &\equiv |
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[2282] | 361 | \sum_{\substack{i_p,\,k_p}} |
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| 362 | \begin{pmatrix} |
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| 363 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ |
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| 364 | \\ |
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| 365 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) \\ |
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| 366 | \end{pmatrix}. |
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| 367 | \end{flalign} |
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[3294] | 368 | \subsection{Ensuring the scheme does not increase tracer variance} |
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| 369 | \label{sec:triad:variance} |
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[2282] | 370 | |
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[3294] | 371 | We now require that this operator should not increase the |
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[2282] | 372 | globally-integrated tracer variance. |
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| 373 | %This changes according to |
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| 374 | % \begin{align*} |
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| 375 | % &\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\ |
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[3294] | 376 | % &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
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| 377 | % \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
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[2282] | 378 | % + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\ |
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[3294] | 379 | % &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
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[2282] | 380 | % {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] |
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| 381 | % + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\ |
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| 382 | % \end{align*} |
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| 383 | Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ drives a lateral flux |
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[3294] | 384 | $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ across the $u$-point $i+i_p,k$ and |
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[2282] | 385 | a vertical flux $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ across the |
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[3294] | 386 | $w$-point $i,k+k_p$. The lateral flux drives a net rate of change of |
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| 387 | variance, summed over the two $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of |
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[2282] | 388 | \begin{multline} |
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| 389 | {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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| 390 | \quad {b_T}_{i+i_p+1/2}^k\left(\frac{\partial T}{\partial |
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| 391 | t}T\right)_{i+i_p+1/2}^k \\ |
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| 392 | \begin{split} |
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| 393 | &= -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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| 394 | {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\ |
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[3294] | 395 | &={\;} _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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[2282] | 396 | \end{split} |
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| 397 | \end{multline} |
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[3294] | 398 | while the vertical flux similarly drives a net rate of change of |
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| 399 | variance summed over the $T$-points $i,k+k_p-\half$ (above) and |
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| 400 | $i,k+k_p+\half$ (below) of |
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[2282] | 401 | \begin{equation} |
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[3294] | 402 | \label{eq:triad:dvar_iso_k} |
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[2282] | 403 | _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. |
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| 404 | \end{equation} |
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| 405 | The total variance tendency driven by the triad is the sum of these |
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| 406 | two. Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and |
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[3294] | 407 | $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with \eqref{eq:triad:latflux-triad} and |
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| 408 | \eqref{eq:triad:vertflux-triad}, it is |
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[2282] | 409 | \begin{multline*} |
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[3294] | 410 | -\Alts_i^k\left \{ |
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[2282] | 411 | { } _i^k{\mathbb{A}_u}_{i_p}^{k_p} |
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| 412 | \left( |
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| 413 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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| 414 | - {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
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| 415 | \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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| 416 | - \left. { } _i^k{\mathbb{A}_w}_{i_p}^{k_p} |
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| 417 | \left( |
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| 418 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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| 419 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
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| 420 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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| 421 | \right) {\,}_i^k\mathbb{R}_{i_p}^{k_p}\delta_{k+ k_p}[T^i] |
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| 422 | \right \}. |
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| 423 | \end{multline*} |
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| 424 | The key point is then that if we require |
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| 425 | $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$ and $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$ |
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| 426 | to be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by |
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| 427 | \begin{equation} |
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[3294] | 428 | \label{eq:triad:V-A} |
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[2282] | 429 | _i^k\mathbb{V}_{i_p}^{k_p} |
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| 430 | ={\;}_i^k{\mathbb{A}_u}_{i_p}^{k_p}\,{e_{1u}}_{\,i + i_p}^{\,k} |
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| 431 | ={\;}_i^k{\mathbb{A}_w}_{i_p}^{k_p}\,{e_{3w}}_{\,i}^{\,k + k_p}, |
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| 432 | \end{equation} |
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| 433 | the variance tendency reduces to the perfect square |
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| 434 | \begin{equation} |
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[3294] | 435 | \label{eq:triad:perfect-square} |
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| 436 | -\Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} |
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[2282] | 437 | \left( |
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| 438 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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| 439 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
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| 440 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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| 441 | \right)^2\leq 0. |
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| 442 | \end{equation} |
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[3294] | 443 | 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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[2282] | 444 | with a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase |
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| 445 | the net variance. Since the total fluxes are sums of such fluxes from |
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| 446 | the various triads, this constraint, applied to all triads, is |
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| 447 | sufficient to ensure that the globally integrated variance does not |
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| 448 | increase. |
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| 449 | |
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[3294] | 450 | The expression \eqref{eq:triad:V-A} can be interpreted as a discretization |
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[2282] | 451 | of the global integral |
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| 452 | \begin{equation} |
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[3294] | 453 | \label{eq:triad:cts-var} |
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| 454 | \frac{\partial}{\partial t}\int\!\half T^2\, dV = |
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| 455 | \int\!\mathbf{F}\cdot\nabla T\, dV, |
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[2282] | 456 | \end{equation} |
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| 457 | where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the |
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| 458 | lateral and vertical fluxes/unit area |
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| 459 | \[ |
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| 460 | \mathbf{F}=\left( |
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[3294] | 461 | \left.{}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_u}_{i_p}^{k_p}, |
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| 462 | \left.{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_w}_{i_p}^{k_p} |
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[2282] | 463 | \right) |
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| 464 | \] |
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| 465 | and the gradient |
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| 466 | \[\nabla T = \left( |
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[3294] | 467 | \left.\delta_{i+ i_p}[T^k] \right/ {e_{1u}}_{\,i + i_p}^{\,k}, |
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| 468 | \left.\delta_{k+ k_p}[T^i] \right/ {e_{3w}}_{\,i}^{\,k + k_p} |
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[2282] | 469 | \right) |
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| 470 | \] |
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| 471 | \subsection{Triad volumes in Griffes's scheme and in \NEMO} |
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| 472 | To complete the discretization we now need only specify the triad |
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| 473 | volumes $_i^k\mathbb{V}_{i_p}^{k_p}$. \citet{Griffies_al_JPO98} identify |
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| 474 | these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter |
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[3294] | 475 | cells, defined in terms of the distances between $T$, $u$,$f$ and |
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| 476 | $w$-points. This is the natural discretization of |
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| 477 | \eqref{eq:triad:cts-var}. The \NEMO model, however, operates with scale |
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[2282] | 478 | factors instead of grid sizes, and scale factors for the quarter |
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| 479 | cells are not defined. Instead, therefore we simply choose |
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| 480 | \begin{equation} |
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[3294] | 481 | \label{eq:triad:V-NEMO} |
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[2282] | 482 | _i^k\mathbb{V}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k, |
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| 483 | \end{equation} |
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[3294] | 484 | as a quarter of the volume of the $u$-cell inside which the triad |
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[2282] | 485 | quarter-cell lies. This has the nice property that when the slopes |
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| 486 | $\mathbb{R}$ vanish, the lateral flux from tracer cell $i,k$ to |
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| 487 | $i+1,k$ reduces to the classical form |
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| 488 | \begin{equation} |
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[3294] | 489 | \label{eq:triad:lat-normal} |
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| 490 | -\overline\Alts_{\,i+1/2}^k\; |
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[2282] | 491 | \frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} |
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| 492 | \;\frac{\delta_{i+ 1/2}[T^k] }{{e_{1u}}_{\,i + i_p}^{\,k}} |
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[3294] | 493 | = -\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}}. |
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[2282] | 494 | \end{equation} |
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[3294] | 495 | In fact if the diffusive coefficient is defined at $u$-points, so that |
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| 496 | we employ $\Alts_{i+i_p}^k$ instead of $\Alts_i^k$ in the definitions of the |
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| 497 | triad fluxes \eqref{eq:triad:latflux-triad} and \eqref{eq:triad:vertflux-triad}, |
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[2282] | 498 | we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above. |
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| 499 | |
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| 500 | \subsection{Summary of the scheme} |
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[3294] | 501 | The iso-neutral fluxes at $u$- and |
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| 502 | $w$-points are the sums of the triad fluxes that cross the $u$- and |
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| 503 | $w$-faces \eqref{Eq_iso_flux}: |
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| 504 | \begin{subequations}\label{eq:triad:alltriadflux} |
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| 505 | \begin{flalign}\label{eq:triad:vect_isoflux} |
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| 506 | \vect{F}_\mathrm{iso}(T) &\equiv |
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| 507 | \sum_{\substack{i_p,\,k_p}} |
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| 508 | \begin{pmatrix} |
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| 509 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ |
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| 510 | \\ |
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| 511 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
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| 512 | \end{pmatrix}, |
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| 513 | \end{flalign} |
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| 514 | where \eqref{eq:triad:latflux-triad}: |
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| 515 | \begin{align} |
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| 516 | \label{eq:triad:triadfluxu} |
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| 517 | _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - \Alts_i^k{ |
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| 518 | \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}} |
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| 519 | \left( |
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| 520 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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| 521 | -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
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| 522 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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| 523 | \right),\\ |
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| 524 | \intertext{and} |
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| 525 | _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) |
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| 526 | &= \Alts_i^k{\: }\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{3w}}_{\,i}^{\,k+k_p}} |
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| 527 | \left( |
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| 528 | {_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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| 529 | -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ |
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| 530 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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| 531 | \right),\label{eq:triad:triadfluxw} |
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| 532 | \end{align} |
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| 533 | with \eqref{eq:triad:V-NEMO} |
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| 534 | \begin{equation} |
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| 535 | \label{eq:triad:V-NEMO2} |
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| 536 | _i^k{\mathbb{V}}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k. |
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| 537 | \end{equation} |
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| 538 | \end{subequations} |
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| 539 | |
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| 540 | The divergence of the expression \eqref{Eq_iso_flux} for the fluxes gives the iso-neutral diffusion tendency at |
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[2282] | 541 | each tracer point: |
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[3294] | 542 | \begin{equation} \label{eq:triad:iso_operator} D_l^T = \frac{1}{b_T} |
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[2282] | 543 | \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k |
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| 544 | {\mathbb{F}_u }_{i_p}^{k_p}} \right] + \delta_{k} \left[ |
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| 545 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} |
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| 546 | \end{equation} |
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| 547 | where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. |
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| 548 | The diffusion scheme satisfies the following six properties: |
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| 549 | \begin{description} |
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| 550 | \item[$\bullet$ horizontal diffusion] The discretization of the |
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[3294] | 551 | diffusion operator recovers \eqref{eq:triad:lat-normal} the traditional five-point Laplacian in |
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[2282] | 552 | the limit of flat iso-neutral direction : |
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[3294] | 553 | \begin{equation} \label{eq:triad:iso_property0} D_l^T = \frac{1}{b_T} \ |
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[2282] | 554 | \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; |
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[3294] | 555 | \overline\Alts^{\,i} \; \delta_{i+1/2}[T] \right] \qquad |
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[2282] | 556 | \text{when} \quad { _i^k \mathbb{R}_{i_p}^{k_p} }=0 |
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| 557 | \end{equation} |
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| 558 | |
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| 559 | \item[$\bullet$ implicit treatment in the vertical] Only tracer values |
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| 560 | associated with a single water column appear in the expression |
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[3294] | 561 | \eqref{eq:triad:i33} for the $_{33}$ fluxes, vertical fluxes driven by |
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[2282] | 562 | vertical gradients. This is of paramount importance since it means |
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[3294] | 563 | that a time-implicit algorithm can be used to solve the vertical |
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| 564 | diffusion equation. This is necessary |
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| 565 | since the vertical eddy |
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[2282] | 566 | diffusivity associated with this term, |
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| 567 | \begin{equation} |
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[3294] | 568 | \frac{1}{b_w}\sum_{\substack{i_p, \,k_p}} \left\{ |
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| 569 | {\:}_i^k\mathbb{V}_{i_p}^{k_p} \: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
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| 570 | \right\} = |
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| 571 | \frac{1}{4b_w}\sum_{\substack{i_p, \,k_p}} \left\{ |
---|
| 572 | {b_u}_{i+i_p}^k\: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
---|
[2282] | 573 | \right\}, |
---|
| 574 | \end{equation} |
---|
| 575 | (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large. |
---|
| 576 | |
---|
| 577 | \item[$\bullet$ pure iso-neutral operator] The iso-neutral flux of |
---|
| 578 | locally referenced potential density is zero. See |
---|
[3294] | 579 | \eqref{eq:triad:latflux-rho} and \eqref{eq:triad:vertflux-triad2}. |
---|
[2282] | 580 | |
---|
| 581 | \item[$\bullet$ conservation of tracer] The iso-neutral diffusion |
---|
| 582 | conserves tracer content, $i.e.$ |
---|
[3294] | 583 | \begin{equation} \label{eq:triad:iso_property1} \sum_{i,j,k} \left\{ D_l^T \ |
---|
[2282] | 584 | b_T \right\} = 0 |
---|
| 585 | \end{equation} |
---|
| 586 | This property is trivially satisfied since the iso-neutral diffusive |
---|
| 587 | operator is written in flux form. |
---|
| 588 | |
---|
| 589 | \item[$\bullet$ no increase of tracer variance] The iso-neutral diffusion |
---|
| 590 | does not increase the tracer variance, $i.e.$ |
---|
[3294] | 591 | \begin{equation} \label{eq:triad:iso_property2} \sum_{i,j,k} \left\{ T \ D_l^T |
---|
[2282] | 592 | \ b_T \right\} \leq 0 |
---|
| 593 | \end{equation} |
---|
| 594 | The property is demonstrated in |
---|
[3294] | 595 | \S\ref{sec:triad:variance} above. It is a key property for a diffusion |
---|
| 596 | term. It means that it is also a dissipation term, $i.e.$ it |
---|
| 597 | dissipates the square of the quantity on which it is applied. It |
---|
[2282] | 598 | therefore ensures that, when the diffusivity coefficient is large |
---|
[3294] | 599 | enough, the field on which it is applied becomes free of grid-point |
---|
[2282] | 600 | noise. |
---|
| 601 | |
---|
| 602 | \item[$\bullet$ self-adjoint operator] The iso-neutral diffusion |
---|
| 603 | operator is self-adjoint, $i.e.$ |
---|
[3294] | 604 | \begin{equation} \label{eq:triad:iso_property3} \sum_{i,j,k} \left\{ S \ D_l^T |
---|
[2282] | 605 | \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} |
---|
| 606 | \end{equation} |
---|
| 607 | In other word, there is no need to develop a specific routine from |
---|
| 608 | the adjoint of this operator. We just have to apply the same |
---|
| 609 | routine. This property can be demonstrated similarly to the proof of |
---|
| 610 | the `no increase of tracer variance' property. The contribution by a |
---|
[3294] | 611 | single triad towards the left hand side of \eqref{eq:triad:iso_property3}, can |
---|
| 612 | be found by replacing $\delta[T]$ by $\delta[S]$ in \eqref{eq:triad:dvar_iso_i} |
---|
| 613 | and \eqref{eq:triad:dvar_iso_k}. This results in a term similar to |
---|
| 614 | \eqref{eq:triad:perfect-square}, |
---|
[2282] | 615 | \begin{equation} |
---|
[3294] | 616 | \label{eq:triad:TScovar} |
---|
| 617 | - \Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} |
---|
[2282] | 618 | \left( |
---|
| 619 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 620 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
| 621 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
| 622 | \right) |
---|
| 623 | \left( |
---|
| 624 | \frac{ \delta_{i+ i_p}[S^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 625 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
| 626 | \frac{ \delta_{k+k_p} [S^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
| 627 | \right). |
---|
| 628 | \end{equation} |
---|
| 629 | This is symmetrical in $T $ and $S$, so exactly the same term arises |
---|
| 630 | from the discretization of this triad's contribution towards the |
---|
[3294] | 631 | RHS of \eqref{eq:triad:iso_property3}. |
---|
[2282] | 632 | \end{description} |
---|
[3294] | 633 | \subsection{Treatment of the triads at the boundaries}\label{sec:triad:iso_bdry} |
---|
| 634 | The triad slope can only be defined where both the grid boxes centred at |
---|
| 635 | the end of the arms exist. Triads that would poke up |
---|
| 636 | through the upper ocean surface into the atmosphere, or down into the |
---|
| 637 | ocean floor, must be masked out. See Fig.~\ref{fig:triad:bdry_triads}. Surface layer triads |
---|
| 638 | $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and |
---|
| 639 | $\triad{i+1}{1}{R}{-1/2}{-1/2}$ (blue) that require density to be |
---|
| 640 | specified above the ocean surface are masked (Fig.~\ref{fig:triad:bdry_triads}a): this ensures that lateral |
---|
| 641 | tracer gradients produce no flux through the ocean surface. However, |
---|
| 642 | to prevent surface noise, it is customary to retain the $_{11}$ contributions towards |
---|
| 643 | the lateral triad fluxes $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and |
---|
| 644 | $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$; this drives diapycnal tracer |
---|
| 645 | fluxes. Similar comments apply to triads that would intersect the |
---|
| 646 | ocean floor (Fig.~\ref{fig:triad:bdry_triads}b). Note that both near bottom |
---|
| 647 | triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and |
---|
| 648 | $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ |
---|
| 649 | or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point is |
---|
| 650 | masked. The associated lateral fluxes (grey-black dashed line) are |
---|
[6289] | 651 | masked if \np{ln\_botmix\_triad}=false, but left unmasked, |
---|
| 652 | giving bottom mixing, if \np{ln\_botmix\_triad}=true. |
---|
[2282] | 653 | |
---|
[6289] | 654 | The default option \np{ln\_botmix\_triad}=false is suitable when the |
---|
[3294] | 655 | bbl mixing option is enabled (\key{trabbl}, with \np{nn\_bbl\_ldf}=1), |
---|
| 656 | or for simple idealized problems. For setups with topography without |
---|
[6289] | 657 | bbl mixing, \np{ln\_botmix\_triad}=true may be necessary. |
---|
[3294] | 658 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 659 | \begin{figure}[h] \begin{center} |
---|
[3297] | 660 | \includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_GRIFF_bdry_triads} |
---|
[3294] | 661 | \caption{ \label{fig:triad:bdry_triads} |
---|
| 662 | (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer |
---|
| 663 | points (black dots), and $i+1/2,1$ $u$-point (blue square). Triad |
---|
| 664 | slopes $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and $\triad{i+1}{1}{R}{-1/2}{-1/2}$ |
---|
| 665 | (blue) poking through the ocean surface are masked (faded in |
---|
| 666 | figure). However, the lateral $_{11}$ contributions towards |
---|
| 667 | $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$ |
---|
| 668 | (yellow line) are still applied, giving diapycnal diffusive |
---|
| 669 | fluxes.\\ |
---|
| 670 | (b) Both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and |
---|
| 671 | $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ |
---|
| 672 | or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point |
---|
| 673 | is masked. The associated lateral fluxes (grey-black dashed |
---|
[6289] | 674 | line) are masked if \np{botmix\_triad}=.false., but left |
---|
| 675 | unmasked, giving bottom mixing, if \np{botmix\_triad}=.true.} |
---|
[3294] | 676 | \end{center} \end{figure} |
---|
| 677 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 678 | \subsection{ Limiting of the slopes within the interior}\label{sec:triad:limit} |
---|
| 679 | As discussed in \S\ref{LDF_slp_iso}, iso-neutral slopes relative to |
---|
| 680 | geopotentials must be bounded everywhere, both for consistency with the small-slope |
---|
| 681 | approximation and for numerical stability \citep{Cox1987, |
---|
| 682 | Griffies_Bk04}. The bound chosen in \NEMO is applied to each |
---|
| 683 | component of the slope separately and has a value of $1/100$ in the ocean interior. |
---|
| 684 | %, ramping linearly down above 70~m depth to zero at the surface |
---|
| 685 | It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to |
---|
| 686 | geopotentials (here the $\sigma_i$ are the slopes of the coordinate surfaces relative to |
---|
| 687 | geopotentials) \eqref{Eq_PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate |
---|
| 688 | surfaces, so we require |
---|
| 689 | \begin{equation*} |
---|
| 690 | |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01. |
---|
| 691 | \end{equation*} |
---|
| 692 | and then recalculate the slopes $r_i$ relative to coordinates. |
---|
| 693 | Each individual triad slope |
---|
| 694 | \begin{equation} |
---|
| 695 | \label{eq:triad:Rtilde} |
---|
| 696 | _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}} |
---|
| 697 | \end{equation} |
---|
| 698 | is limited like this and then the corresponding |
---|
| 699 | $_i^k\mathbb{R}_{i_p}^{k_p} $ are recalculated and combined to form the fluxes. |
---|
| 700 | Note that where the slopes have been limited, there is now a non-zero |
---|
| 701 | iso-neutral density flux that drives dianeutral mixing. In particular this iso-neutral density flux |
---|
| 702 | is always downwards, and so acts to reduce gravitational potential energy. |
---|
| 703 | \subsection{Tapering within the surface mixed layer}\label{sec:triad:taper} |
---|
[2282] | 704 | |
---|
[3294] | 705 | Additional tapering of the iso-neutral fluxes is necessary within the |
---|
| 706 | surface mixed layer. When the Griffies triads are used, we offer two |
---|
| 707 | options for this. |
---|
| 708 | \subsubsection{Linear slope tapering within the surface mixed layer}\label{sec:triad:lintaper} |
---|
| 709 | This is the option activated by the default choice |
---|
| 710 | \np{ln\_triad\_iso}=false. Slopes $\tilde{r}_i$ relative to |
---|
| 711 | geopotentials are tapered linearly from their value immediately below the mixed layer to zero at the |
---|
| 712 | surface, as described in option (c) of Fig.~\ref{Fig_eiv_slp}, to values |
---|
| 713 | \begin{subequations} |
---|
| 714 | \begin{equation} |
---|
| 715 | \label{eq:triad:rmtilde} |
---|
| 716 | \rMLt = |
---|
| 717 | -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for } z>-h, |
---|
| 718 | \end{equation} |
---|
| 719 | and then the $r_i$ relative to vertical coordinate surfaces are appropriately |
---|
| 720 | adjusted to |
---|
| 721 | \begin{equation} |
---|
| 722 | \label{eq:triad:rm} |
---|
| 723 | \rML =\rMLt -\sigma_i \quad \text{ for } z>-h. |
---|
| 724 | \end{equation} |
---|
| 725 | \end{subequations} |
---|
| 726 | Thus the diffusion operator within the mixed layer is given by: |
---|
| 727 | \begin{equation} \label{eq:triad:iso_tensor_ML} |
---|
| 728 | D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad |
---|
| 729 | \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} |
---|
| 730 | 1 \hfill & 0 \hfill & {-\rML[1]}\hfill \\ |
---|
| 731 | 0 \hfill & 1 \hfill & {-\rML[2]} \hfill \\ |
---|
| 732 | {-\rML[1]}\hfill & {-\rML[2]} \hfill & {\rML[1]^2+\rML[2]^2} \hfill |
---|
| 733 | \end{array} }} \right) |
---|
| 734 | \end{equation} |
---|
| 735 | |
---|
| 736 | This slope tapering gives a natural connection between tracer in the |
---|
| 737 | mixed-layer and in isopycnal layers immediately below, in the |
---|
| 738 | thermocline. It is consistent with the way the $\tilde{r}_i$ are |
---|
| 739 | tapered within the mixed layer (see \S\ref{sec:triad:taperskew} below) |
---|
| 740 | so as to ensure a uniform GM eddy-induced velocity throughout the |
---|
| 741 | mixed layer. However, it gives a downwards density flux and so acts so |
---|
| 742 | as to reduce potential energy in the same way as does the slope |
---|
| 743 | limiting discussed above in \S\ref{sec:triad:limit}. |
---|
| 744 | |
---|
| 745 | As in \S\ref{sec:triad:limit} above, the tapering |
---|
| 746 | \eqref{eq:triad:rmtilde} is applied separately to each triad |
---|
| 747 | $_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the |
---|
| 748 | $_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted. For clarity, we assume |
---|
| 749 | $z$-coordinates in the following; the conversion from |
---|
| 750 | $\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as described |
---|
| 751 | above by \eqref{eq:triad:Rtilde}. |
---|
| 752 | \begin{enumerate} |
---|
| 753 | \item Mixed-layer depth is defined so as to avoid including regions of weak |
---|
| 754 | vertical stratification in the slope definition. |
---|
| 755 | At each $i,j$ (simplified to $i$ in |
---|
| 756 | Fig.~\ref{fig:triad:MLB_triad}), we define the mixed-layer by setting |
---|
| 757 | the vertical index of the tracer point immediately below the mixed |
---|
| 758 | layer, $k_\mathrm{ML}$, as the maximum $k$ (shallowest tracer point) |
---|
| 759 | such that the potential density |
---|
| 760 | ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, where $i,k_{10}$ is |
---|
| 761 | the tracer gridbox within which the depth reaches 10~m. See the left |
---|
| 762 | side of Fig.~\ref{fig:triad:MLB_triad}. We use the $k_{10}$-gridbox |
---|
| 763 | instead of the surface gridbox to avoid problems e.g.\ with thin |
---|
| 764 | daytime mixed-layers. Currently we use the same |
---|
| 765 | $\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is |
---|
| 766 | used to output the diagnosed mixed-layer depth |
---|
| 767 | $h_\mathrm{ML}=|z_{W}|_{k_\mathrm{ML}+1/2}$, the depth of the $w$-point |
---|
| 768 | above the $i,k_\mathrm{ML}$ tracer point. |
---|
| 769 | |
---|
| 770 | \item We define `basal' triad slopes |
---|
| 771 | ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ as the slopes |
---|
| 772 | of those triads whose vertical `arms' go down from the |
---|
| 773 | $i,k_\mathrm{ML}$ tracer point to the $i,k_\mathrm{ML}-1$ tracer point |
---|
| 774 | below. This is to ensure that the vertical density gradients |
---|
| 775 | associated with these basal triad slopes |
---|
| 776 | ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ are |
---|
| 777 | representative of the thermocline. The four basal triads defined in the bottom part |
---|
| 778 | of Fig.~\ref{fig:triad:MLB_triad} are then |
---|
| 779 | \begin{align} |
---|
| 780 | {\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p} &= |
---|
| 781 | {\:}^{k_\mathrm{ML}-k_p-1/2}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}, \label{eq:triad:Rbase} |
---|
| 782 | \\ |
---|
| 783 | \intertext{with e.g.\ the green triad} |
---|
| 784 | {\:}_i{\mathbb{R}_\mathrm{base}}_{1/2}^{-1/2}&= |
---|
| 785 | {\:}^{k_\mathrm{ML}}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{-1/2}. \notag |
---|
| 786 | \end{align} |
---|
| 787 | The vertical flux associated with each of these triads passes through the $w$-point |
---|
| 788 | $i,k_\mathrm{ML}-1/2$ lying \emph{below} the $i,k_\mathrm{ML}$ tracer point, |
---|
| 789 | so it is this depth |
---|
| 790 | \begin{equation} |
---|
| 791 | \label{eq:triad:zbase} |
---|
| 792 | {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2} |
---|
| 793 | \end{equation} |
---|
| 794 | (one gridbox deeper than the |
---|
| 795 | diagnosed ML depth $z_\mathrm{ML})$ that sets the $h$ used to taper |
---|
| 796 | the slopes in \eqref{eq:triad:rmtilde}. |
---|
| 797 | \item Finally, we calculate the adjusted triads |
---|
| 798 | ${\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,i_p}^{k_p}$ within the mixed |
---|
| 799 | layer, by multiplying the appropriate |
---|
| 800 | ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ by the ratio of |
---|
| 801 | the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_\mathrm{base}}_{\,i}$. For |
---|
| 802 | instance the green triad centred on $i,k$ |
---|
| 803 | \begin{align} |
---|
| 804 | {\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,1/2}^{-1/2} &= |
---|
| 805 | \frac{{z_w}_{k-1/2}}{{z_\mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{-1/2} |
---|
| 806 | \notag \\ |
---|
| 807 | \intertext{and more generally} |
---|
| 808 | {\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,i_p}^{k_p} &= |
---|
| 809 | \frac{{z_w}_{k+k_p}}{{z_\mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}.\label{eq:triad:RML} |
---|
| 810 | \end{align} |
---|
| 811 | \end{enumerate} |
---|
| 812 | |
---|
| 813 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 814 | \begin{figure}[h] |
---|
| 815 | \fcapside {\caption{\label{fig:triad:MLB_triad} Definition of |
---|
| 816 | mixed-layer depth and calculation of linearly tapered |
---|
| 817 | triads. The figure shows a water column at a given $i,j$ |
---|
| 818 | (simplified to $i$), with the ocean surface at the top. Tracer points are denoted by |
---|
| 819 | bullets, and black lines the edges of the tracer cells; $k$ |
---|
| 820 | increases upwards. \\ |
---|
| 821 | \hspace{5 em}We define the mixed-layer by setting the vertical index |
---|
| 822 | of the tracer point immediately below the mixed layer, |
---|
| 823 | $k_\mathrm{ML}$, as the maximum $k$ (shallowest tracer point) |
---|
| 824 | such that ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, |
---|
| 825 | where $i,k_{10}$ is the tracer gridbox within which the depth |
---|
| 826 | reaches 10~m. We calculate the triad slopes within the mixed |
---|
| 827 | layer by linearly tapering them from zero (at the surface) to |
---|
| 828 | the `basal' slopes, the slopes of the four triads passing through the |
---|
| 829 | $w$-point $i,k_\mathrm{ML}-1/2$ (blue square), |
---|
| 830 | ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$. Triads with |
---|
| 831 | different $i_p,k_p$, denoted by different colours, (e.g. the green |
---|
| 832 | triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.}} |
---|
[3297] | 833 | {\includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_GRIFF_MLB_triads}} |
---|
[3294] | 834 | \end{figure} |
---|
| 835 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 836 | |
---|
| 837 | \subsubsection{Additional truncation of skew iso-neutral flux |
---|
| 838 | components} |
---|
| 839 | \label{sec:triad:Gerdes-taper} |
---|
| 840 | The alternative option is activated by setting \np{ln\_triad\_iso} = |
---|
| 841 | true. This retains the same tapered slope $\rML$ described above for the |
---|
| 842 | calculation of the $_{33}$ term of the iso-neutral diffusion tensor (the |
---|
| 843 | vertical tracer flux driven by vertical tracer gradients), but |
---|
| 844 | replaces the $\rML$ in the skew term by |
---|
| 845 | \begin{equation} |
---|
| 846 | \label{eq:triad:rm*} |
---|
| 847 | \rML^*=\left.\rMLt^2\right/\tilde{r}_i-\sigma_i, |
---|
| 848 | \end{equation} |
---|
| 849 | giving a ML diffusive operator |
---|
| 850 | \begin{equation} \label{eq:triad:iso_tensor_ML2} |
---|
| 851 | D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad |
---|
| 852 | \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} |
---|
| 853 | 1 \hfill & 0 \hfill & {-\rML[1]^*}\hfill \\ |
---|
| 854 | 0 \hfill & 1 \hfill & {-\rML[2]^*} \hfill \\ |
---|
| 855 | {-\rML[1]^*}\hfill & {-\rML[2]^*} \hfill & {\rML[1]^2+\rML[2]^2} \hfill \\ |
---|
| 856 | \end{array} }} \right). |
---|
| 857 | \end{equation} |
---|
| 858 | This operator |
---|
| 859 | \footnote{To ensure good behaviour where horizontal density |
---|
| 860 | gradients are weak, we in fact follow \citet{Gerdes1991} and set |
---|
| 861 | $\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$.} |
---|
| 862 | then has the property it gives no vertical density flux, and so does |
---|
| 863 | not change the potential energy. |
---|
| 864 | This approach is similar to multiplying the iso-neutral diffusion |
---|
| 865 | coefficient by $\tilde{r}_\mathrm{max}^{-2}\tilde{r}_i^{-2}$ for steep |
---|
| 866 | slopes, as suggested by \citet{Gerdes1991} (see also \citet{Griffies_Bk04}). |
---|
| 867 | Again it is applied separately to each triad $_i^k\mathbb{R}_{i_p}^{k_p}$ |
---|
| 868 | |
---|
| 869 | In practice, this approach gives weak vertical tracer fluxes through |
---|
| 870 | the mixed-layer, as well as vanishing density fluxes. While it is |
---|
| 871 | theoretically advantageous that it does not change the potential |
---|
| 872 | energy, it may give a discontinuity between the |
---|
| 873 | fluxes within the mixed-layer (purely horizontal) and just below (along |
---|
| 874 | iso-neutral surfaces). |
---|
| 875 | % This may give strange looking results, |
---|
| 876 | % particularly where the mixed-layer depth varies strongly laterally. |
---|
[2282] | 877 | % ================================================================ |
---|
| 878 | % Skew flux formulation for Eddy Induced Velocity : |
---|
| 879 | % ================================================================ |
---|
[3294] | 880 | \section{Eddy induced advection formulated as a skew flux}\label{sec:triad:skew-flux} |
---|
[2282] | 881 | |
---|
[3294] | 882 | \subsection{The continuous skew flux formulation}\label{sec:triad:continuous-skew-flux} |
---|
| 883 | |
---|
| 884 | When Gent and McWilliams's [1990] diffusion is used, |
---|
| 885 | an additional advection term is added. The associated velocity is the so called |
---|
[2282] | 886 | eddy induced velocity, the formulation of which depends on the slopes of iso- |
---|
[3294] | 887 | neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used |
---|
| 888 | here are referenced to the geopotential surfaces, $i.e.$ \eqref{Eq_ldfslp_geo} |
---|
[2282] | 889 | is used in $z$-coordinate, and the sum \eqref{Eq_ldfslp_geo} |
---|
[3294] | 890 | + \eqref{Eq_ldfslp_iso} in $z^*$ or $s$-coordinates. |
---|
[2282] | 891 | |
---|
[3294] | 892 | The eddy induced velocity is given by: |
---|
| 893 | \begin{subequations} \label{eq:triad:eiv} |
---|
| 894 | \begin{equation}\label{eq:triad:eiv_v} |
---|
[2282] | 895 | \begin{split} |
---|
[3294] | 896 | u^* & = - \frac{1}{e_{3}}\; \partial_i\psi_1, \\ |
---|
| 897 | v^* & = - \frac{1}{e_{3}}\; \partial_j\psi_2, \\ |
---|
| 898 | w^* & = \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i \left( e_{2} \, \psi_1\right) |
---|
| 899 | + \partial_j \left( e_{1} \, \psi_2\right) \right\}, |
---|
[2282] | 900 | \end{split} |
---|
| 901 | \end{equation} |
---|
[3294] | 902 | where the streamfunctions $\psi_i$ are given by |
---|
| 903 | \begin{equation} \label{eq:triad:eiv_psi} |
---|
| 904 | \begin{split} |
---|
| 905 | \psi_1 & = A_{e} \; \tilde{r}_1, \\ |
---|
| 906 | \psi_2 & = A_{e} \; \tilde{r}_2, |
---|
| 907 | \end{split} |
---|
| 908 | \end{equation} |
---|
| 909 | \end{subequations} |
---|
| 910 | 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] | 911 | |
---|
[3294] | 912 | The traditional way to implement this additional advection is to add |
---|
| 913 | it to the Eulerian velocity prior to computing the tracer |
---|
| 914 | advection. This is implemented if \key{traldf\_eiv} is set in the |
---|
[6289] | 915 | default implementation, where \np{ln\_traldf\_triad} is set |
---|
[3294] | 916 | false. This allows us to take advantage of all the advection schemes |
---|
| 917 | offered for the tracers (see \S\ref{TRA_adv}) and not just a $2^{nd}$ |
---|
| 918 | order advection scheme. This is particularly useful for passive |
---|
| 919 | tracers where \emph{positivity} of the advection scheme is of |
---|
| 920 | paramount importance. |
---|
[2282] | 921 | |
---|
[6289] | 922 | However, when \np{ln\_traldf\_triad} is set true, \NEMO instead |
---|
[3294] | 923 | implements eddy induced advection according to the so-called skew form |
---|
| 924 | \citep{Griffies_JPO98}. It is based on a transformation of the advective fluxes |
---|
| 925 | using the non-divergent nature of the eddy induced velocity. |
---|
| 926 | For example in the (\textbf{i},\textbf{k}) plane, the tracer advective |
---|
| 927 | fluxes per unit area in $ijk$ space can be |
---|
[2282] | 928 | transformed as follows: |
---|
| 929 | \begin{flalign*} |
---|
| 930 | \begin{split} |
---|
[3294] | 931 | \textbf{F}_\mathrm{eiv}^T = |
---|
| 932 | \begin{pmatrix} |
---|
[2282] | 933 | {e_{2}\,e_{3}\; u^*} \\ |
---|
| 934 | {e_{1}\,e_{2}\; w^*} \\ |
---|
| 935 | \end{pmatrix} \; T |
---|
| 936 | &= |
---|
[3294] | 937 | \begin{pmatrix} |
---|
| 938 | { - \partial_k \left( e_{2} \,\psi_1 \right) \; T \;} \\ |
---|
| 939 | {+ \partial_i \left( e_{2} \, \psi_1 \right) \; T \;} \\ |
---|
[2282] | 940 | \end{pmatrix} \\ |
---|
[3294] | 941 | &= |
---|
| 942 | \begin{pmatrix} |
---|
| 943 | { - \partial_k \left( e_{2} \, \psi_1 \; T \right) \;} \\ |
---|
| 944 | {+ \partial_i \left( e_{2} \,\psi_1 \; T \right) \;} \\ |
---|
| 945 | \end{pmatrix} |
---|
| 946 | + |
---|
| 947 | \begin{pmatrix} |
---|
| 948 | {+ e_{2} \, \psi_1 \; \partial_k T} \\ |
---|
| 949 | { - e_{2} \, \psi_1 \; \partial_i T} \\ |
---|
| 950 | \end{pmatrix} |
---|
[2282] | 951 | \end{split} |
---|
| 952 | \end{flalign*} |
---|
[3294] | 953 | and since the eddy induced velocity field is non-divergent, we end up with the skew |
---|
| 954 | form of the eddy induced advective fluxes per unit area in $ijk$ space: |
---|
| 955 | \begin{equation} \label{eq:triad:eiv_skew_ijk} |
---|
| 956 | \textbf{F}_\mathrm{eiv}^T = \begin{pmatrix} |
---|
| 957 | {+ e_{2} \, \psi_1 \; \partial_k T} \\ |
---|
| 958 | { - e_{2} \, \psi_1 \; \partial_i T} \\ |
---|
[2282] | 959 | \end{pmatrix} |
---|
| 960 | \end{equation} |
---|
[3294] | 961 | The total fluxes per unit physical area are then |
---|
| 962 | \begin{equation}\label{eq:triad:eiv_skew_physical} |
---|
| 963 | \begin{split} |
---|
| 964 | f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T \\ |
---|
| 965 | f^*_2 & = \frac{1}{e_{3}}\; \psi_2 \partial_k T \\ |
---|
| 966 | f^*_3 & = -\frac{1}{e_{1}e_{2}}\; \left\{ e_{2} \psi_1 \partial_i T |
---|
| 967 | + e_{1} \psi_2 \partial_j T \right\}. \\ |
---|
| 968 | \end{split} |
---|
| 969 | \end{equation} |
---|
| 970 | Note that Eq.~ \eqref{eq:triad:eiv_skew_physical} takes the same form whatever the |
---|
| 971 | vertical coordinate, though of course the slopes |
---|
| 972 | $\tilde{r}_i$ which define the $\psi_i$ in \eqref{eq:triad:eiv_psi} are relative to geopotentials. |
---|
| 973 | The tendency associated with eddy induced velocity is then simply the convergence |
---|
| 974 | of the fluxes (\ref{eq:triad:eiv_skew_ijk}, \ref{eq:triad:eiv_skew_physical}), so |
---|
| 975 | \begin{equation} \label{eq:triad:skew_eiv_conv} |
---|
| 976 | \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 } \left[ |
---|
| 977 | \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right) |
---|
| 978 | + \frac{\partial}{\partial j} \left( e_1 \; |
---|
| 979 | \psi_2 \partial_k T\right) |
---|
| 980 | - \frac{\partial}{\partial k} \left( e_{2} \psi_1 \partial_i T |
---|
| 981 | + e_{1} \psi_2 \partial_j T \right) \right] |
---|
| 982 | \end{equation} |
---|
| 983 | It naturally conserves the tracer content, as it is expressed in flux |
---|
| 984 | form. Since it has the same divergence as the advective form it also |
---|
| 985 | preserves the tracer variance. |
---|
[2282] | 986 | |
---|
[3294] | 987 | \subsection{The discrete skew flux formulation} |
---|
| 988 | The skew fluxes in (\ref{eq:triad:eiv_skew_physical}, \ref{eq:triad:eiv_skew_ijk}), like the off-diagonal terms |
---|
| 989 | (\ref{eq:triad:i13c}, \ref{eq:triad:i31c}) of the small angle diffusion tensor, are best |
---|
| 990 | expressed in terms of the triad slopes, as in Fig.~\ref{fig:triad:ISO_triad} |
---|
| 991 | and Eqs~(\ref{eq:triad:i13}, \ref{eq:triad:i31}); but now in terms of the triad slopes |
---|
| 992 | $\tilde{\mathbb{R}}$ relative to geopotentials instead of the |
---|
| 993 | $\mathbb{R}$ relative to coordinate surfaces. The discrete form of |
---|
| 994 | \eqref{eq:triad:eiv_skew_ijk} using the slopes \eqref{eq:triad:R} and |
---|
| 995 | defining $A_e$ at $T$-points is then given by: |
---|
[2282] | 996 | |
---|
| 997 | |
---|
[3294] | 998 | \begin{subequations}\label{eq:triad:allskewflux} |
---|
| 999 | \begin{flalign}\label{eq:triad:vect_skew_flux} |
---|
| 1000 | \vect{F}_\mathrm{eiv}(T) &\equiv |
---|
| 1001 | \sum_{\substack{i_p,\,k_p}} |
---|
| 1002 | \begin{pmatrix} |
---|
| 1003 | {_{i+1/2-i_p}^k {\mathbb{S}_u}_{i_p}^{k_p} } (T) \\ |
---|
| 1004 | \\ |
---|
| 1005 | {_i^{k+1/2-k_p} {\mathbb{S}_w}_{i_p}^{k_p} } (T) \\ |
---|
| 1006 | \end{pmatrix}, |
---|
| 1007 | \end{flalign} |
---|
| 1008 | where the skew flux in the $i$-direction associated with a given |
---|
| 1009 | triad is (\ref{eq:triad:latflux-triad}, \ref{eq:triad:triadfluxu}): |
---|
| 1010 | \begin{align} |
---|
| 1011 | \label{eq:triad:skewfluxu} |
---|
| 1012 | _i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) &= + \quarter {A_e}_i^k{ |
---|
| 1013 | \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
| 1014 | \ {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} \ |
---|
| 1015 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }, |
---|
| 1016 | \\ |
---|
| 1017 | \intertext{and \eqref{eq:triad:triadfluxw} in the $k$-direction, changing the sign |
---|
| 1018 | to be consistent with \eqref{eq:triad:eiv_skew_ijk}:} |
---|
| 1019 | _i^k {\mathbb{S}_w}_{i_p}^{k_p} (T) |
---|
| 1020 | &= -\quarter {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}} |
---|
| 1021 | {_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} |
---|
| 1022 | \end{align} |
---|
| 1023 | \end{subequations} |
---|
[2282] | 1024 | |
---|
[3294] | 1025 | Such a discretisation is consistent with the iso-neutral |
---|
| 1026 | operator as it uses the same definition for the slopes. It also |
---|
| 1027 | ensures the following two key properties. |
---|
| 1028 | \subsubsection{No change in tracer variance} |
---|
| 1029 | The discretization conserves tracer variance, $i.e.$ it does not |
---|
| 1030 | include a diffusive component but is a `pure' advection term. This can |
---|
| 1031 | be seen |
---|
| 1032 | %either from Appendix \ref{Apdx_eiv_skew} or |
---|
| 1033 | by considering the |
---|
| 1034 | fluxes associated with a given triad slope |
---|
| 1035 | $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. For, following |
---|
| 1036 | \S\ref{sec:triad:variance} and \eqref{eq:triad:dvar_iso_i}, the |
---|
| 1037 | associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$ |
---|
| 1038 | drives a net rate of change of variance, summed over the two |
---|
| 1039 | $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of |
---|
| 1040 | \begin{equation} |
---|
| 1041 | \label{eq:triad:dvar_eiv_i} |
---|
| 1042 | _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], |
---|
| 1043 | \end{equation} |
---|
| 1044 | while the associated vertical skew-flux gives a variance change summed over the |
---|
| 1045 | $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of |
---|
| 1046 | \begin{equation} |
---|
| 1047 | \label{eq:triad:dvar_eiv_k} |
---|
| 1048 | _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. |
---|
| 1049 | \end{equation} |
---|
| 1050 | Inspection of the definitions (\ref{eq:triad:skewfluxu}, \ref{eq:triad:skewfluxw}) |
---|
| 1051 | shows that these two variance changes (\ref{eq:triad:dvar_eiv_i}, \ref{eq:triad:dvar_eiv_k}) |
---|
| 1052 | sum to zero. Hence the two fluxes associated with each triad make no |
---|
| 1053 | net contribution to the variance budget. |
---|
[2282] | 1054 | |
---|
[3294] | 1055 | \subsubsection{Reduction in gravitational PE} |
---|
| 1056 | The vertical density flux associated with the vertical skew-flux |
---|
| 1057 | always has the same sign as the vertical density gradient; thus, so |
---|
| 1058 | long as the fluid is stable (the vertical density gradient is |
---|
| 1059 | negative) the vertical density flux is negative (downward) and hence |
---|
| 1060 | reduces the gravitational PE. |
---|
[2282] | 1061 | |
---|
[3294] | 1062 | For the change in gravitational PE driven by the $k$-flux is |
---|
| 1063 | \begin{align} |
---|
| 1064 | \label{eq:triad:vert_densityPE} |
---|
| 1065 | g {e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) |
---|
| 1066 | &=g {e_{3w}}_{\,i}^{\,k+k_p}\left[-\alpha _i^k {\:}_i^k |
---|
| 1067 | {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k |
---|
| 1068 | {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\ |
---|
| 1069 | \intertext{Substituting ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from |
---|
| 1070 | \eqref{eq:triad:skewfluxw}, gives} |
---|
| 1071 | % and separating out |
---|
| 1072 | % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$, |
---|
| 1073 | % gives two terms. The |
---|
| 1074 | % first $\rtriad{R}$ term (the only term for $z$-coordinates) is: |
---|
| 1075 | &=-\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} |
---|
| 1076 | \frac{ -\alpha _i^k\delta_{i+ i_p}[T^k]+ \beta_i^k\delta_{i+ i_p}[S^k]} { {e_{1u}}_{\,i + i_p}^{\,k} } \notag \\ |
---|
| 1077 | &=+\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
| 1078 | \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) {_i^k\mathbb{R}_{i_p}^{k_p}} |
---|
| 1079 | \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}, |
---|
| 1080 | \end{align} |
---|
| 1081 | using the definition of the triad slope $\rtriad{R}$, |
---|
| 1082 | \eqref{eq:triad:R} to express $-\alpha _i^k\delta_{i+ i_p}[T^k]+ |
---|
| 1083 | \beta_i^k\delta_{i+ i_p}[S^k]$ in terms of $-\alpha_i^k \delta_{k+ |
---|
| 1084 | k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$. |
---|
[2282] | 1085 | |
---|
[3294] | 1086 | Where the coordinates slope, the $i$-flux gives a PE change |
---|
| 1087 | \begin{multline} |
---|
| 1088 | \label{eq:triad:lat_densityPE} |
---|
| 1089 | g \delta_{i+i_p}[z_T^k] |
---|
| 1090 | \left[ |
---|
| 1091 | -\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) |
---|
| 1092 | \right] \\ |
---|
| 1093 | = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
| 1094 | \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
| 1095 | \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) |
---|
| 1096 | \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}}, |
---|
| 1097 | \end{multline} |
---|
| 1098 | (using \eqref{eq:triad:skewfluxu}) and so the total PE change |
---|
| 1099 | \eqref{eq:triad:vert_densityPE} + \eqref{eq:triad:lat_densityPE} associated with the triad fluxes is |
---|
| 1100 | \begin{multline} |
---|
| 1101 | \label{eq:triad:tot_densityPE} |
---|
| 1102 | g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) + |
---|
| 1103 | g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\ |
---|
| 1104 | = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
| 1105 | \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 |
---|
| 1106 | \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}}. |
---|
| 1107 | \end{multline} |
---|
| 1108 | Where the fluid is stable, with $-\alpha_i^k \delta_{k+ k_p}[T^i]+ |
---|
| 1109 | \beta_i^k\delta_{k+ k_p}[S^i]<0$, this PE change is negative. |
---|
[2282] | 1110 | |
---|
[3294] | 1111 | \subsection{Treatment of the triads at the boundaries}\label{sec:triad:skew_bdry} |
---|
| 1112 | Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes |
---|
| 1113 | are masked at the boundaries in exactly the same way as are the triad |
---|
| 1114 | slopes \rtriad{R} used for the iso-neutral diffusive fluxes, as |
---|
| 1115 | described in \S\ref{sec:triad:iso_bdry} and |
---|
| 1116 | Fig.~\ref{fig:triad:bdry_triads}. Thus surface layer triads |
---|
| 1117 | $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are |
---|
| 1118 | masked, and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ |
---|
| 1119 | and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the |
---|
| 1120 | $i,k+1$ or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ |
---|
[6289] | 1121 | $u$-point is masked. The namelist parameter \np{ln\_botmix\_triad} has |
---|
[3294] | 1122 | no effect on the eddy-induced skew-fluxes. |
---|
[2282] | 1123 | |
---|
[3294] | 1124 | \subsection{ Limiting of the slopes within the interior}\label{sec:triad:limitskew} |
---|
| 1125 | Presently, the iso-neutral slopes $\tilde{r}_i$ relative |
---|
| 1126 | to geopotentials are limited to be less than $1/100$, exactly as in |
---|
| 1127 | calculating the iso-neutral diffusion, \S \ref{sec:triad:limit}. Each |
---|
| 1128 | individual triad \rtriadt{R} is so limited. |
---|
[2282] | 1129 | |
---|
[3294] | 1130 | \subsection{Tapering within the surface mixed layer}\label{sec:triad:taperskew} |
---|
| 1131 | The slopes $\tilde{r}_i$ relative to |
---|
| 1132 | geopotentials (and thus the individual triads \rtriadt{R}) are always tapered linearly from their value immediately below the mixed layer to zero at the |
---|
| 1133 | surface \eqref{eq:triad:rmtilde}, as described in \S\ref{sec:triad:lintaper}. This is |
---|
| 1134 | option (c) of Fig.~\ref{Fig_eiv_slp}. This linear tapering for the |
---|
| 1135 | slopes used to calculate the eddy-induced fluxes is |
---|
| 1136 | unaffected by the value of \np{ln\_triad\_iso}. |
---|
[2282] | 1137 | |
---|
[3294] | 1138 | The justification for this linear slope tapering is that, for $A_e$ |
---|
| 1139 | that is constant or varies only in the horizontal (the most commonly |
---|
| 1140 | used options in \NEMO: see \S\ref{LDF_coef}), it is |
---|
| 1141 | equivalent to a horizontal eiv (eddy-induced velocity) that is uniform |
---|
| 1142 | within the mixed layer \eqref{eq:triad:eiv_v}. This ensures that the |
---|
| 1143 | eiv velocities do not restratify the mixed layer \citep{Treguier1997, |
---|
| 1144 | Danabasoglu_al_2008}. Equivantly, in terms |
---|
| 1145 | of the skew-flux formulation we use here, the |
---|
| 1146 | linear slope tapering within the mixed-layer gives a linearly varying |
---|
| 1147 | vertical flux, and so a tracer convergence uniform in depth (the |
---|
| 1148 | horizontal flux convergence is relatively insignificant within the mixed-layer). |
---|
| 1149 | |
---|
| 1150 | \subsection{Streamfunction diagnostics}\label{sec:triad:sfdiag} |
---|
| 1151 | Where the namelist parameter \np{ln\_traldf\_gdia}=true, diagnosed |
---|
| 1152 | mean eddy-induced velocities are output. Each time step, |
---|
| 1153 | streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at |
---|
| 1154 | $uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$ |
---|
| 1155 | (integer $i$, integer +1/2 $j$, integer +1/2 $k$) points (see Table |
---|
| 1156 | \ref{Tab_cell}) respectively. We follow \citep{Griffies_Bk04} and |
---|
| 1157 | calculate the streamfunction at a given $uw$-point from the |
---|
| 1158 | surrounding four triads according to: |
---|
| 1159 | \begin{equation} |
---|
| 1160 | \label{eq:triad:sfdiagi} |
---|
| 1161 | {\psi_1}_{i+1/2}^{k+1/2}={\quarter}\sum_{\substack{i_p,\,k_p}} |
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| 1162 | {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}. |
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| 1163 | \end{equation} |
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| 1164 | The streamfunction $\psi_1$ is calculated similarly at $vw$ points. |
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| 1165 | The eddy-induced velocities are then calculated from the |
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| 1166 | straightforward discretisation of \eqref{eq:triad:eiv_v}: |
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| 1167 | \begin{equation}\label{eq:triad:eiv_v_discrete} |
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| 1168 | \begin{split} |
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| 1169 | {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), \\ |
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| 1170 | {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), \\ |
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| 1171 | {w^*}_{i,j}^{k+1/2} & = \frac{1}{e_{1t}e_{2t}}\; \left\{ |
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| 1172 | {e_{2u}}_{i+1/2}^{k+1/2} \,{\psi_1}_{i+1/2}^{k+1/2} - |
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| 1173 | {e_{2u}}_{i-1/2}^{k+1/2} \,{\psi_1}_{i-1/2}^{k+1/2} \right. + \\ |
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| 1174 | \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\}, |
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| 1175 | \end{split} |
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| 1176 | \end{equation} |
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