[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
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| 2 | |
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[6997] | 3 | \begin{document} |
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[2282] | 4 | % ================================================================ |
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[3294] | 5 | % Iso-neutral diffusion : |
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[2282] | 6 | % ================================================================ |
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[9407] | 7 | \chapter[Iso-Neutral Diffusion and Eddy Advection using Triads] |
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| 8 | {\texorpdfstring{Iso-Neutral Diffusion and\\ Eddy Advection using Triads}{Iso-Neutral Diffusion and Eddy Advection using Triads}} |
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| 9 | \label{apdx:triad} |
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[10414] | 10 | |
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[2282] | 11 | \minitoc |
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[10414] | 12 | |
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| 13 | \newpage |
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| 14 | |
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[9364] | 15 | \section{Choice of \protect\ngn{namtra\_ldf} namelist parameters} |
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[3294] | 16 | %-----------------------------------------nam_traldf------------------------------------------------------ |
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[10146] | 17 | |
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| 18 | \nlst{namtra_ldf} |
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[3294] | 19 | %--------------------------------------------------------------------------------------------------------- |
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[2282] | 20 | |
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[10354] | 21 | Two scheme are available to perform the iso-neutral diffusion. |
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| 22 | If the namelist logical \np{ln\_traldf\_triad} is set true, |
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| 23 | \NEMO updates both active and passive tracers using the Griffies triad representation of iso-neutral diffusion and |
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| 24 | the eddy-induced advective skew (GM) fluxes. |
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| 25 | If the namelist logical \np{ln\_traldf\_iso} is set true, |
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| 26 | the filtered version of Cox's original scheme (the Standard scheme) is employed (\autoref{sec:LDF_slp}). |
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| 27 | In the present implementation of the Griffies scheme, |
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[9393] | 28 | the advective skew fluxes are implemented even if \np{ln\_traldf\_eiv} is false. |
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[6289] | 29 | |
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[10354] | 30 | Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}. |
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| 31 | Note that when GM fluxes are used, the eddy-advective (GM) velocities are output for diagnostic purposes using xIOS, |
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[6289] | 32 | even though the eddy advection is accomplished by means of the skew fluxes. |
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[2282] | 33 | |
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[6289] | 34 | |
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[3294] | 35 | The options specific to the Griffies scheme include: |
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| 36 | \begin{description}[font=\normalfont] |
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[10354] | 37 | \item[\np{ln\_triad\_iso}] |
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| 38 | See \autoref{sec:taper}. |
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| 39 | If this is set false (the default), |
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| 40 | then `iso-neutral' mixing is accomplished within the surface mixed-layer along slopes linearly decreasing with |
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| 41 | depth from the value immediately below the mixed-layer to zero (flat) at the surface (\autoref{sec:lintaper}). |
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| 42 | This is the same treatment as used in the default implementation |
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| 43 | \autoref{subsec:LDF_slp_iso}; \autoref{fig:eiv_slp}. |
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| 44 | Where \np{ln\_triad\_iso} is set true, |
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| 45 | the vertical skew flux is further reduced to ensure no vertical buoyancy flux, |
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| 46 | giving an almost pure horizontal diffusive tracer flux within the mixed layer. |
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| 47 | This is similar to the tapering suggested by \citet{Gerdes1991}. See \autoref{subsec:Gerdes-taper} |
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| 48 | \item[\np{ln\_botmix\_triad}] |
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| 49 | See \autoref{sec:iso_bdry}. |
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[6289] | 50 | If this is set false (the default) then the lateral diffusive fluxes |
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| 51 | associated with triads partly masked by topography are neglected. |
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| 52 | If it is set true, however, then these lateral diffusive fluxes are applied, |
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| 53 | giving smoother bottom tracer fields at the cost of introducing diapycnal mixing. |
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[10354] | 54 | \item[\np{rn\_sw\_triad}] |
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| 55 | blah blah to be added.... |
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[3294] | 56 | \end{description} |
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[6289] | 57 | The options shared with the Standard scheme include: |
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| 58 | \begin{description}[font=\normalfont] |
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[9393] | 59 | \item[\np{ln\_traldf\_msc}] blah blah to be added |
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| 60 | \item[\np{rn\_slpmax}] blah blah to be added |
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[6289] | 61 | \end{description} |
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[9393] | 62 | |
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[3294] | 63 | \section{Triad formulation of iso-neutral diffusion} |
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[9407] | 64 | \label{sec:iso} |
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[10414] | 65 | |
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[10354] | 66 | We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98}, |
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[6289] | 67 | but formulated within the \NEMO framework, using scale factors rather than grid-sizes. |
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[2282] | 68 | |
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[9393] | 69 | \subsection{Iso-neutral diffusion operator} |
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[10414] | 70 | |
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[10354] | 71 | The iso-neutral second order tracer diffusive operator for small angles between |
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| 72 | iso-neutral surfaces and geopotentials is given by \autoref{eq:iso_tensor_1}: |
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[10414] | 73 | \begin{subequations} |
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| 74 | \label{eq:iso_tensor_1} |
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[3294] | 75 | \begin{equation} |
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| 76 | D^{lT}=-\Div\vect{f}^{lT}\equiv |
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| 77 | -\frac{1}{e_1e_2e_3}\left[\pd{i}\left (f_1^{lT}e_2e_3\right) + |
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| 78 | \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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| 79 | \end{equation} |
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| 80 | where the diffusive flux per unit area of physical space |
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| 81 | \begin{equation} |
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| 82 | \vect{f}^{lT}=-\Alt\Re\cdot\grad T, |
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| 83 | \end{equation} |
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| 84 | \begin{equation} |
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[9414] | 85 | \label{eq:iso_tensor_2} |
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[3294] | 86 | \mbox{with}\quad \;\;\Re = |
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| 87 | \begin{pmatrix} |
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[10414] | 88 | 1 & 0 & -r_1 \mystrut \\ |
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| 89 | 0 & 1 & -r_2 \mystrut \\ |
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[6289] | 90 | -r_1 & -r_2 & r_1 ^2+r_2 ^2 \mystrut |
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[3294] | 91 | \end{pmatrix} |
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| 92 | \quad \text{and} \quad\grad T= |
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| 93 | \begin{pmatrix} |
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[6289] | 94 | \frac{1}{e_1} \pd[T]{i} \mystrut \\ |
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| 95 | \frac{1}{e_2} \pd[T]{j} \mystrut \\ |
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| 96 | \frac{1}{e_3} \pd[T]{k} \mystrut |
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[10414] | 97 | \end{pmatrix} |
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| 98 | . |
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[3294] | 99 | \end{equation} |
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| 100 | \end{subequations} |
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| 101 | % \left( {{\begin{array}{*{20}c} |
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| 102 | % 1 \hfill & 0 \hfill & {-r_1 } \hfill \\ |
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| 103 | % 0 \hfill & 1 \hfill & {-r_2 } \hfill \\ |
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| 104 | % {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ |
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| 105 | % \end{array} }} \right) |
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[10354] | 106 | Here \autoref{eq:PE_iso_slopes} |
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[2282] | 107 | \begin{align*} |
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| 108 | r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i} |
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[10414] | 109 | \right) |
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| 110 | \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \\ |
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| 111 | &=-\frac{e_3 }{e_1 } \left( -\alpha\frac{\partial T }{\partial i} + |
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| 112 | \beta\frac{\partial S }{\partial i} \right) \left( |
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| 113 | -\alpha\frac{\partial T }{\partial k} + \beta\frac{\partial S |
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| 114 | }{\partial k} \right)^{-1} |
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[2282] | 115 | \end{align*} |
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[10354] | 116 | is the $i$-component of the slope of the iso-neutral surface relative to the computational surface, |
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| 117 | and $r_2$ is the $j$-component. |
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[2282] | 118 | |
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[10354] | 119 | We will find it useful to consider the fluxes per unit area in $i,j,k$ space; we write |
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[10414] | 120 | \[ |
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| 121 | % \label{eq:Fijk} |
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[3294] | 122 | \vect{F}_{\mathrm{iso}}=\left(f_1^{lT}e_2e_3, f_2^{lT}e_1e_3, f_3^{lT}e_1e_2\right). |
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[10414] | 123 | \] |
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[10354] | 124 | Additionally, we will sometimes write the contributions towards the fluxes $\vect{f}$ and |
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| 125 | $\vect{F}_{\mathrm{iso}}$ from the component $R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$, |
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| 126 | with $f_{ij}=R_{ij}e_i^{-1}\partial T/\partial x_i$ (no summation) etc. |
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[3294] | 127 | |
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| 128 | The off-diagonal terms of the small angle diffusion tensor |
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[10354] | 129 | \autoref{eq:iso_tensor_1}, \autoref{eq:iso_tensor_2} produce skew-fluxes along |
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| 130 | the $i$- and $j$-directions resulting from the vertical tracer gradient: |
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[3294] | 131 | \begin{align} |
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[9407] | 132 | \label{eq:i13c} |
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[3294] | 133 | 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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[10414] | 134 | \intertext{and in the k-direction resulting from the lateral tracer gradients} |
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[9407] | 135 | \label{eq:i31c} |
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[10414] | 136 | 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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[3294] | 137 | \end{align} |
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| 138 | |
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[10354] | 139 | The vertical diffusive flux associated with the $_{33}$ component of the small angle diffusion tensor is |
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[2282] | 140 | \begin{equation} |
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[9407] | 141 | \label{eq:i33c} |
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[3294] | 142 | f_{33}=-\Alt(r_1^2 +r_2^2) \frac{1}{e_3}\frac{\partial T}{\partial k}. |
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[2282] | 143 | \end{equation} |
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| 144 | |
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[10354] | 145 | Since there are no cross terms involving $r_1$ and $r_2$ in the above, |
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| 146 | we can consider the iso-neutral diffusive fluxes separately in the $i$-$k$ and $j$-$k$ planes, |
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| 147 | just adding together the vertical components from each plane. |
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| 148 | The following description will describe the fluxes on the $i$-$k$ plane. |
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[2282] | 149 | |
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[10354] | 150 | There is no natural discretization for the $i$-component of the skew-flux, \autoref{eq:i13c}, |
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| 151 | as although it must be evaluated at $u$-points, |
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| 152 | it involves vertical gradients (both for the tracer and the slope $r_1$), defined at $w$-points. |
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| 153 | Similarly, the vertical skew flux, \autoref{eq:i31c}, |
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| 154 | is evaluated at $w$-points but involves horizontal gradients defined at $u$-points. |
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[2282] | 155 | |
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[9393] | 156 | \subsection{Standard discretization} |
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[10414] | 157 | |
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[2282] | 158 | The straightforward approach to discretize the lateral skew flux |
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[10354] | 159 | \autoref{eq:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 into OPA, |
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| 160 | \autoref{eq:tra_ldf_iso}, is to calculate a mean vertical gradient at the $u$-point from |
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| 161 | the average of the four surrounding vertical tracer gradients, and multiply this by a mean slope at the $u$-point, |
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| 162 | calculated from the averaged surrounding vertical density gradients. |
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| 163 | The total area-integrated skew-flux (flux per unit area in $ijk$ space) from tracer cell $i,k$ to $i+1,k$, |
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| 164 | noting that the $e_{{3}_{i+1/2}^k}$ in the area $e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with |
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| 165 | the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer gradient, is then \autoref{eq:tra_ldf_iso} |
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[10406] | 166 | \[ |
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[3294] | 167 | \left(F_u^{13} \right)_{i+\hhalf}^k = \Alts_{i+\hhalf}^k |
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| 168 | {e_{2}}_{i+1/2}^k \overline{\overline |
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[2282] | 169 | r_1} ^{\,i,k}\,\overline{\overline{\delta_k T}}^{\,i,k}, |
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[10406] | 170 | \] |
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[2282] | 171 | where |
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[10406] | 172 | \[ |
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[2282] | 173 | \overline{\overline |
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[10414] | 174 | r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k} |
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[3294] | 175 | \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}}, |
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[10406] | 176 | \] |
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[10354] | 177 | and here and in the following we drop the $^{lT}$ superscript from $\Alt$ for simplicity. |
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| 178 | Unfortunately the resulting combination $\overline{\overline{\delta_k\bullet}}^{\,i,k}$ of a $k$ average and |
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| 179 | a $k$ difference of the tracer reduces to $\bullet_{k+1}-\bullet_{k-1}$, |
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| 180 | so two-grid-point oscillations are invisible to this discretization of the iso-neutral operator. |
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| 181 | These \emph{computational modes} will not be damped by this operator, and may even possibly be amplified by it. |
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| 182 | Consequently, applying this operator to a tracer does not guarantee the decrease of its global-average variance. |
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| 183 | To correct this, we introduced a smoothing of the slopes of the iso-neutral surfaces (see \autoref{chap:LDF}). |
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| 184 | This technique works for $T$ and $S$ in so far as they are active tracers |
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| 185 | ($i.e.$ they enter the computation of density), but it does not work for a passive tracer. |
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[9393] | 186 | |
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[2282] | 187 | \subsection{Expression of the skew-flux in terms of triad slopes} |
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[10414] | 188 | |
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[10354] | 189 | \citep{Griffies_al_JPO98} introduce a different discretization of the off-diagonal terms that |
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| 190 | nicely solves the problem. |
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[3294] | 191 | % Instead of multiplying the mean slope calculated at the $u$-point by |
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| 192 | % the mean vertical gradient at the $u$-point, |
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[2282] | 193 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[10414] | 194 | \begin{figure}[tb] |
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| 195 | \begin{center} |
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[6997] | 196 | \includegraphics[width=1.05\textwidth]{Fig_GRIFF_triad_fluxes} |
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[10414] | 197 | \caption{ |
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| 198 | \protect\label{fig:ISO_triad} |
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[2376] | 199 | (a) Arrangement of triads $S_i$ and tracer gradients to |
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[10414] | 200 | give lateral tracer flux from box $i,k$ to $i+1,k$ |
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[2376] | 201 | (b) Triads $S'_i$ and tracer gradients to give vertical tracer flux from |
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[10414] | 202 | box $i,k$ to $i,k+1$. |
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| 203 | } |
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| 204 | \end{center} |
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| 205 | \end{figure} |
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[2282] | 206 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[10354] | 207 | They get the skew flux from the products of the vertical gradients at each $w$-point surrounding the $u$-point with |
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| 208 | the corresponding `triad' slope calculated from the lateral density gradient across the $u$-point divided by |
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| 209 | the vertical density gradient at the same $w$-point as the tracer gradient. |
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| 210 | See \autoref{fig:ISO_triad}a, where the thick lines denote the tracer gradients, |
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| 211 | and the thin lines the corresponding triads, with slopes $s_1, \dotsc s_4$. |
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| 212 | The total area-integrated skew-flux from tracer cell $i,k$ to $i+1,k$ |
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[2282] | 213 | \begin{multline} |
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[9407] | 214 | \label{eq:i13} |
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[3294] | 215 | \left( F_u^{13} \right)_{i+\frac{1}{2}}^k = \Alts_{i+1}^k a_1 s_1 |
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[10406] | 216 | \delta_{k+\frac{1}{2}} \left[ T^{i+1} |
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[3294] | 217 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} + \Alts _i^k a_2 s_2 \delta |
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[2282] | 218 | _{k+\frac{1}{2}} \left[ T^i |
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| 219 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} \\ |
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[10414] | 220 | +\Alts _{i+1}^k a_3 s_3 \delta_{k-\frac{1}{2}} \left[ T^{i+1} |
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[3294] | 221 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} +\Alts _i^k a_4 s_4 \delta |
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[2282] | 222 | _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}, |
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| 223 | \end{multline} |
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[10354] | 224 | where the contributions of the triad fluxes are weighted by areas $a_1, \dotsc a_4$, |
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| 225 | and $\Alts$ is now defined at the tracer points rather than the $u$-points. |
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| 226 | This discretization gives a much closer stencil, and disallows the two-point computational modes. |
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[2282] | 227 | |
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[10354] | 228 | The vertical skew flux \autoref{eq:i31c} from tracer cell $i,k$ to $i,k+1$ at |
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| 229 | the $w$-point $i,k+\hhalf$ is constructed similarly (\autoref{fig:ISO_triad}b) by |
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| 230 | multiplying lateral tracer gradients from each of the four surrounding $u$-points by the appropriate triad slope: |
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[2282] | 231 | \begin{multline} |
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[9407] | 232 | \label{eq:i31} |
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[3294] | 233 | \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} = \Alts_i^{k+1} a_{1}' |
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[10406] | 234 | s_{1}' \delta_{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1} |
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[10414] | 235 | +\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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[10406] | 236 | + \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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| 237 | +\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] | 238 | \end{multline} |
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[3294] | 239 | |
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| 240 | We notate the triad slopes $s_i$ and $s'_i$ in terms of the `anchor point' $i,k$ |
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[10354] | 241 | (appearing in both the vertical and lateral gradient), |
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| 242 | and the $u$- and $w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the triad as follows |
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| 243 | (see also \autoref{fig:ISO_triad}): |
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[2282] | 244 | \begin{equation} |
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[9407] | 245 | \label{eq:R} |
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[3294] | 246 | _i^k \mathbb{R}_{i_p}^{k_p} |
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| 247 | =-\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} |
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[2282] | 248 | \ |
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[3294] | 249 | \frac |
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[6289] | 250 | { \alpha_i^k \ \delta_{i+i_p}[T^k] - \beta_i^k \ \delta_{i+i_p}[S^k] } |
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| 251 | { \alpha_i^k \ \delta_{k+k_p}[T^i] - \beta_i^k \ \delta_{k+k_p}[S^i] }. |
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[2282] | 252 | \end{equation} |
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[10354] | 253 | In calculating the slopes of the local neutral surfaces, |
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| 254 | the expansion coefficients $\alpha$ and $\beta$ are evaluated at the anchor points of the triad, |
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[6289] | 255 | while the metrics are calculated at the $u$- and $w$-points on the arms. |
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[2282] | 256 | |
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| 257 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[10414] | 258 | \begin{figure}[tb] |
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| 259 | \begin{center} |
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[6997] | 260 | \includegraphics[width=0.80\textwidth]{Fig_GRIFF_qcells} |
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[10414] | 261 | \caption{ |
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| 262 | \protect\label{fig:qcells} |
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[10354] | 263 | Triad notation for quarter cells. $T$-cells are inside boxes, |
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| 264 | while the $i+\half,k$ $u$-cell is shaded in green and |
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[10414] | 265 | the $i,k+\half$ $w$-cell is shaded in pink. |
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| 266 | } |
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| 267 | \end{center} |
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| 268 | \end{figure} |
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[2282] | 269 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 270 | |
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[10354] | 271 | Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (\autoref{fig:qcells}) with the quarter cell that is |
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| 272 | 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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| 273 | Expressing the slopes $s_i$ and $s'_i$ in \autoref{eq:i13} and \autoref{eq:i31} in this notation, |
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| 274 | we have $e.g.$ \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. |
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| 275 | Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to |
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| 276 | calculate the lateral flux along its $u$-arm, at $(i+i_p,k)$, |
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| 277 | and then again as an $s'$ to calculate the vertical flux along its $w$-arm at $(i,k+k_p)$. |
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| 278 | Each vertical area $a_i$ used to calculate the lateral flux and horizontal area $a'_i$ used to |
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| 279 | calculate the vertical flux can also be identified as the area across the $u$- and $w$-arms of a unique triad, |
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| 280 | and we notate these areas, similarly to the triad slopes, |
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| 281 | as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, |
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| 282 | where $e.g.$ in \autoref{eq:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, |
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[9407] | 283 | and in \autoref{eq:i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. |
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[2282] | 284 | |
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[9393] | 285 | \subsection{Full triad fluxes} |
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[10414] | 286 | |
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[10354] | 287 | A key property of iso-neutral diffusion is that it should not affect the (locally referenced) density. |
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| 288 | In particular there should be no lateral or vertical density flux. |
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| 289 | The lateral density flux disappears so long as the area-integrated lateral diffusive flux from |
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| 290 | tracer cell $i,k$ to $i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the form |
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[2282] | 291 | \begin{equation} |
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[9407] | 292 | \label{eq:i11} |
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[2282] | 293 | \left( F_u^{11} \right) _{i+\frac{1}{2}} ^{k} = |
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[3294] | 294 | - \left( \Alts_i^{k+1} a_{1} + \Alts_i^{k+1} a_{2} + \Alts_i^k |
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| 295 | a_{3} + \Alts_i^k a_{4} \right) |
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[10406] | 296 | \frac{\delta_{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}}, |
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[2282] | 297 | \end{equation} |
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[10354] | 298 | where the areas $a_i$ are as in \autoref{eq:i13}. |
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| 299 | In this case, separating the total lateral flux, the sum of \autoref{eq:i13} and \autoref{eq:i11}, |
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| 300 | into triad components, a lateral tracer flux |
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[2282] | 301 | \begin{equation} |
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[9407] | 302 | \label{eq:latflux-triad} |
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[3294] | 303 | _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] | 304 | \left( |
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| 305 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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| 306 | -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
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| 307 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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| 308 | \right) |
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| 309 | \end{equation} |
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[10354] | 310 | can be identified with each triad. |
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| 311 | Then, because the same metric factors ${e_{3w}}_{\,i}^{\,k+k_p}$ and ${e_{1u}}_{\,i+i_p}^{\,k}$ are employed for both |
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| 312 | the density gradients in $ _i^k \mathbb{R}_{i_p}^{k_p}$ and the tracer gradients, |
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| 313 | the lateral density flux associated with each triad separately disappears. |
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[2282] | 314 | \begin{equation} |
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[9407] | 315 | \label{eq:latflux-rho} |
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[2282] | 316 | {\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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| 317 | \end{equation} |
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[10354] | 318 | Thus the total flux $\left( F_u^{31} \right) ^i _{i,k+\frac{1}{2}} + \left( F_u^{11} \right) ^i _{i,k+\frac{1}{2}}$ from |
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| 319 | tracer cell $i,k$ to $i+1,k$ must also vanish since it is a sum of four such triad fluxes. |
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[2282] | 320 | |
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[10354] | 321 | The squared slope $r_1^2$ in the expression \autoref{eq:i33c} for the $_{33}$ component is also expressed in |
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| 322 | terms of area-weighted squared triad slopes, |
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| 323 | so the area-integrated vertical flux from tracer cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is |
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[2282] | 324 | \begin{equation} |
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[9407] | 325 | \label{eq:i33} |
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[2282] | 326 | \left( F_w^{33} \right) _i^{k+\frac{1}{2}} = |
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[10414] | 327 | - \left( \Alts_i^{k+1} a_{1}' s_{1}'^2 |
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[3294] | 328 | + \Alts_i^{k+1} a_{2}' s_{2}'^2 |
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| 329 | + \Alts_i^k a_{3}' s_{3}'^2 |
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| 330 | + \Alts_i^k a_{4}' s_{4}'^2 \right)\delta_{k+\frac{1}{2}} \left[ T^{i+1} \right], |
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[2282] | 331 | \end{equation} |
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[10354] | 332 | where the areas $a'$ and slopes $s'$ are the same as in \autoref{eq:i31}. |
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| 333 | Then, separating the total vertical flux, the sum of \autoref{eq:i31} and \autoref{eq:i33}, |
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| 334 | into triad components, a vertical flux |
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[2282] | 335 | \begin{align} |
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[9407] | 336 | \label{eq:vertflux-triad} |
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[2282] | 337 | _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) |
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[3294] | 338 | &= \Alts_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p} |
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[10414] | 339 | \left( |
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[2282] | 340 | {_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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| 341 | -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ |
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| 342 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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[10414] | 343 | \right) \\ |
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[2282] | 344 | &= - \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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[10414] | 345 | {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:vertflux-triad2} |
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[2282] | 346 | \end{align} |
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[10354] | 347 | may be associated with each triad. |
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| 348 | Each vertical density flux $_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)$ associated with a triad then |
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| 349 | separately disappears (because the lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (\rho)$ disappears). |
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| 350 | Consequently the total vertical density flux |
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| 351 | $\left( F_w^{31} \right)_i ^{k+\frac{1}{2}} + \left( F_w^{33} \right)_i^{k+\frac{1}{2}}$ from |
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| 352 | tracer cell $i,k$ to $i,k+1$ must also vanish since it is a sum of four such triad fluxes. |
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[2282] | 353 | |
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[10354] | 354 | We can explicitly identify (\autoref{fig:qcells}) the triads associated with the $s_i$, $a_i$, |
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| 355 | and $s'_i$, $a'_i$ used in the definition of the $u$-fluxes and $w$-fluxes in \autoref{eq:i31}, |
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| 356 | \autoref{eq:i13}, \autoref{eq:i11} \autoref{eq:i33} and \autoref{fig:ISO_triad} to write out |
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| 357 | the iso-neutral fluxes at $u$- and $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces: |
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[9407] | 358 | %(\autoref{fig:ISO_triad}): |
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[10414] | 359 | \begin{flalign} |
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| 360 | \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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[10414] | 363 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\ |
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| 364 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) \\ |
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[2282] | 365 | \end{pmatrix}. |
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| 366 | \end{flalign} |
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[9393] | 367 | |
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[3294] | 368 | \subsection{Ensuring the scheme does not increase tracer variance} |
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[9407] | 369 | \label{subsec:variance} |
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[2282] | 370 | |
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[10354] | 371 | We now require that this operator should not increase the globally-integrated tracer variance. |
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[2282] | 372 | %This changes according to |
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| 373 | % \begin{align*} |
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| 374 | % &\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\ |
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[3294] | 375 | % &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
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| 376 | % \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
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[2282] | 377 | % + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\ |
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[3294] | 378 | % &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
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[2282] | 379 | % {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] |
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| 380 | % + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\ |
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| 381 | % \end{align*} |
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[10354] | 382 | Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ drives a lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ across |
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| 383 | the $u$-point $i+i_p,k$ and a vertical flux $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ across the $w$-point $i,k+k_p$. |
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| 384 | The lateral flux drives a net rate of change of variance, |
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| 385 | summed over the two $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of |
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[2282] | 386 | \begin{multline} |
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| 387 | {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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| 388 | \quad {b_T}_{i+i_p+1/2}^k\left(\frac{\partial T}{\partial |
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| 389 | t}T\right)_{i+i_p+1/2}^k \\ |
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[10414] | 390 | \begin{aligned} |
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| 391 | &= -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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| 392 | {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\ |
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| 393 | &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:dvar_iso_i} |
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| 394 | \end{aligned} |
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[2282] | 395 | \end{multline} |
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[10354] | 396 | while the vertical flux similarly drives a net rate of change of variance summed over |
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| 397 | the $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of |
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[2282] | 398 | \begin{equation} |
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[10414] | 399 | \label{eq:dvar_iso_k} |
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[2282] | 400 | _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. |
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| 401 | \end{equation} |
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[10354] | 402 | The total variance tendency driven by the triad is the sum of these two. |
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| 403 | Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with |
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| 404 | \autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad}, it is |
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[2282] | 405 | \begin{multline*} |
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[10414] | 406 | -\Alts_i^k\left \{ |
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| 407 | { } _i^k{\mathbb{A}_u}_{i_p}^{k_p} |
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| 408 | \left( |
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| 409 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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| 410 | - {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
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| 411 | \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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| 412 | - \left. { } _i^k{\mathbb{A}_w}_{i_p}^{k_p} |
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| 413 | \left( |
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| 414 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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| 415 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
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| 416 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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| 417 | \right) {\,}_i^k\mathbb{R}_{i_p}^{k_p}\delta_{k+ k_p}[T^i] |
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| 418 | \right \}. |
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[2282] | 419 | \end{multline*} |
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[10354] | 420 | The key point is then that if we require $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$ and $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$ to |
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| 421 | be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by |
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[2282] | 422 | \begin{equation} |
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[9407] | 423 | \label{eq:V-A} |
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[2282] | 424 | _i^k\mathbb{V}_{i_p}^{k_p} |
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| 425 | ={\;}_i^k{\mathbb{A}_u}_{i_p}^{k_p}\,{e_{1u}}_{\,i + i_p}^{\,k} |
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| 426 | ={\;}_i^k{\mathbb{A}_w}_{i_p}^{k_p}\,{e_{3w}}_{\,i}^{\,k + k_p}, |
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| 427 | \end{equation} |
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| 428 | the variance tendency reduces to the perfect square |
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| 429 | \begin{equation} |
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[9407] | 430 | \label{eq:perfect-square} |
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[3294] | 431 | -\Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} |
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[2282] | 432 | \left( |
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| 433 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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| 434 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
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| 435 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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| 436 | \right)^2\leq 0. |
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| 437 | \end{equation} |
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[10354] | 438 | Thus, the constraint \autoref{eq:V-A} ensures that the fluxes |
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| 439 | (\autoref{eq:latflux-triad}, \autoref{eq:vertflux-triad}) associated with |
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| 440 | a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase the net variance. |
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| 441 | Since the total fluxes are sums of such fluxes from the various triads, this constraint, applied to all triads, |
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| 442 | is sufficient to ensure that the globally integrated variance does not increase. |
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[2282] | 443 | |
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[10354] | 444 | The expression \autoref{eq:V-A} can be interpreted as a discretization of the global integral |
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[2282] | 445 | \begin{equation} |
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[9407] | 446 | \label{eq:cts-var} |
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[3294] | 447 | \frac{\partial}{\partial t}\int\!\half T^2\, dV = |
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| 448 | \int\!\mathbf{F}\cdot\nabla T\, dV, |
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[2282] | 449 | \end{equation} |
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[10354] | 450 | where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the lateral and vertical fluxes/unit area |
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[2282] | 451 | \[ |
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[10414] | 452 | \mathbf{F}=\left( |
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| 453 | \left.{}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_u}_{i_p}^{k_p}, |
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| 454 | \left.{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_w}_{i_p}^{k_p} |
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| 455 | \right) |
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[2282] | 456 | \] |
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| 457 | and the gradient |
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[10414] | 458 | \[ |
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| 459 | \nabla T = \left( |
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| 460 | \left.\delta_{i+ i_p}[T^k] \right/ {e_{1u}}_{\,i + i_p}^{\,k}, |
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| 461 | \left.\delta_{k+ k_p}[T^i] \right/ {e_{3w}}_{\,i}^{\,k + k_p} |
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| 462 | \right) |
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[2282] | 463 | \] |
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[9393] | 464 | |
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[2282] | 465 | \subsection{Triad volumes in Griffes's scheme and in \NEMO} |
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[10414] | 466 | |
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[10354] | 467 | To complete the discretization we now need only specify the triad volumes $_i^k\mathbb{V}_{i_p}^{k_p}$. |
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| 468 | \citet{Griffies_al_JPO98} identifies these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter cells, |
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| 469 | defined in terms of the distances between $T$, $u$,$f$ and $w$-points. |
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| 470 | This is the natural discretization of \autoref{eq:cts-var}. |
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| 471 | The \NEMO model, however, operates with scale factors instead of grid sizes, |
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| 472 | and scale factors for the quarter cells are not defined. |
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| 473 | Instead, therefore we simply choose |
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[2282] | 474 | \begin{equation} |
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[9407] | 475 | \label{eq:V-NEMO} |
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[2282] | 476 | _i^k\mathbb{V}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k, |
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| 477 | \end{equation} |
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[10354] | 478 | as a quarter of the volume of the $u$-cell inside which the triad quarter-cell lies. |
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| 479 | This has the nice property that when the slopes $\mathbb{R}$ vanish, |
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| 480 | the lateral flux from tracer cell $i,k$ to $i+1,k$ reduces to the classical form |
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[2282] | 481 | \begin{equation} |
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[9407] | 482 | \label{eq:lat-normal} |
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[10414] | 483 | -\overline\Alts_{\,i+1/2}^k\; |
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| 484 | \frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} |
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| 485 | \;\frac{\delta_{i+ 1/2}[T^k] }{{e_{1u}}_{\,i + i_p}^{\,k}} |
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| 486 | = -\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] | 487 | \end{equation} |
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[10354] | 488 | In fact if the diffusive coefficient is defined at $u$-points, |
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| 489 | so that we employ $\Alts_{i+i_p}^k$ instead of $\Alts_i^k$ in the definitions of the triad fluxes |
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| 490 | \autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad}, |
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[2282] | 491 | we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above. |
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| 492 | |
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| 493 | \subsection{Summary of the scheme} |
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[10414] | 494 | |
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[10354] | 495 | The iso-neutral fluxes at $u$- and $w$-points are the sums of the triad fluxes that |
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| 496 | cross the $u$- and $w$-faces \autoref{eq:iso_flux}: |
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[10414] | 497 | \begin{subequations} |
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| 498 | % \label{eq:alltriadflux} |
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| 499 | \begin{flalign*} |
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| 500 | % \label{eq:vect_isoflux} |
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[9364] | 501 | \vect{F}_{\mathrm{iso}}(T) &\equiv |
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[3294] | 502 | \sum_{\substack{i_p,\,k_p}} |
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| 503 | \begin{pmatrix} |
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[10414] | 504 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\ |
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| 505 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
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[3294] | 506 | \end{pmatrix}, |
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[10414] | 507 | \end{flalign*} |
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[9407] | 508 | where \autoref{eq:latflux-triad}: |
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[3294] | 509 | \begin{align} |
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[9407] | 510 | \label{eq:triadfluxu} |
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[3294] | 511 | _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - \Alts_i^k{ |
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[10414] | 512 | \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}} |
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| 513 | \left( |
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| 514 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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| 515 | -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
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| 516 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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| 517 | \right),\\ |
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[3294] | 518 | \intertext{and} |
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| 519 | _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) |
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[10414] | 520 | &= \Alts_i^k{\: }\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{3w}}_{\,i}^{\,k+k_p}} |
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| 521 | \left( |
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| 522 | {_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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| 523 | -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ |
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| 524 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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| 525 | \right),\label{eq:triadfluxw} |
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[3294] | 526 | \end{align} |
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[9407] | 527 | with \autoref{eq:V-NEMO} |
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[10414] | 528 | \[ |
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| 529 | % \label{eq:V-NEMO2} |
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[3294] | 530 | _i^k{\mathbb{V}}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k. |
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[10414] | 531 | \] |
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[3294] | 532 | \end{subequations} |
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| 533 | |
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[10354] | 534 | The divergence of the expression \autoref{eq:iso_flux} for the fluxes gives the iso-neutral diffusion tendency at |
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[2282] | 535 | each tracer point: |
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[10414] | 536 | \[ |
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| 537 | % \label{eq:iso_operator} |
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| 538 | D_l^T = \frac{1}{b_T} |
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[2282] | 539 | \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k |
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| 540 | {\mathbb{F}_u }_{i_p}^{k_p}} \right] + \delta_{k} \left[ |
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| 541 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} |
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[10414] | 542 | \] |
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[2282] | 543 | where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. |
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| 544 | The diffusion scheme satisfies the following six properties: |
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| 545 | \begin{description} |
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[10354] | 546 | \item[$\bullet$ horizontal diffusion] |
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| 547 | The discretization of the diffusion operator recovers the traditional five-point Laplacian |
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| 548 | \autoref{eq:lat-normal} in the limit of flat iso-neutral direction: |
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[10414] | 549 | \[ |
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| 550 | % \label{eq:iso_property0} |
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| 551 | D_l^T = \frac{1}{b_T} \ |
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[2282] | 552 | \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; |
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[3294] | 553 | \overline\Alts^{\,i} \; \delta_{i+1/2}[T] \right] \qquad |
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[2282] | 554 | \text{when} \quad { _i^k \mathbb{R}_{i_p}^{k_p} }=0 |
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[10414] | 555 | \] |
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[2282] | 556 | |
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[10354] | 557 | \item[$\bullet$ implicit treatment in the vertical] |
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| 558 | Only tracer values associated with a single water column appear in the expression \autoref{eq:i33} for |
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| 559 | the $_{33}$ fluxes, vertical fluxes driven by vertical gradients. |
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| 560 | This is of paramount importance since it means that a time-implicit algorithm can be used to |
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| 561 | solve the vertical diffusion equation. |
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| 562 | This is necessary since the vertical eddy diffusivity associated with this term, |
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[10414] | 563 | \[ |
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[3294] | 564 | \frac{1}{b_w}\sum_{\substack{i_p, \,k_p}} \left\{ |
---|
| 565 | {\:}_i^k\mathbb{V}_{i_p}^{k_p} \: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
---|
| 566 | \right\} = |
---|
| 567 | \frac{1}{4b_w}\sum_{\substack{i_p, \,k_p}} \left\{ |
---|
| 568 | {b_u}_{i+i_p}^k\: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
---|
[2282] | 569 | \right\}, |
---|
[10414] | 570 | \] |
---|
[2282] | 571 | (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large. |
---|
| 572 | |
---|
[10354] | 573 | \item[$\bullet$ pure iso-neutral operator] |
---|
| 574 | The iso-neutral flux of locally referenced potential density is zero. |
---|
| 575 | See \autoref{eq:latflux-rho} and \autoref{eq:vertflux-triad2}. |
---|
[2282] | 576 | |
---|
[10354] | 577 | \item[$\bullet$ conservation of tracer] |
---|
| 578 | The iso-neutral diffusion conserves tracer content, $i.e.$ |
---|
[10414] | 579 | \[ |
---|
| 580 | % \label{eq:iso_property1} |
---|
| 581 | \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 |
---|
| 582 | \] |
---|
[10354] | 583 | This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form. |
---|
[2282] | 584 | |
---|
[10354] | 585 | \item[$\bullet$ no increase of tracer variance] |
---|
| 586 | The iso-neutral diffusion does not increase the tracer variance, $i.e.$ |
---|
[10414] | 587 | \[ |
---|
| 588 | % \label{eq:iso_property2} |
---|
| 589 | \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 |
---|
| 590 | \] |
---|
[10354] | 591 | The property is demonstrated in \autoref{subsec:variance} above. |
---|
| 592 | It is a key property for a diffusion term. |
---|
| 593 | It means that it is also a dissipation term, |
---|
| 594 | $i.e.$ it dissipates the square of the quantity on which it is applied. |
---|
| 595 | It therefore ensures that, when the diffusivity coefficient is large enough, |
---|
| 596 | the field on which it is applied becomes free of grid-point noise. |
---|
[2282] | 597 | |
---|
[10354] | 598 | \item[$\bullet$ self-adjoint operator] |
---|
| 599 | The iso-neutral diffusion operator is self-adjoint, $i.e.$ |
---|
[10414] | 600 | \begin{equation} |
---|
| 601 | \label{eq:iso_property3} |
---|
| 602 | \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} |
---|
[2282] | 603 | \end{equation} |
---|
[10354] | 604 | In other word, there is no need to develop a specific routine from the adjoint of this operator. |
---|
| 605 | We just have to apply the same routine. |
---|
| 606 | This property can be demonstrated similarly to the proof of the `no increase of tracer variance' property. |
---|
| 607 | The contribution by a single triad towards the left hand side of \autoref{eq:iso_property3}, |
---|
| 608 | can be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq:dvar_iso_i} and \autoref{eq:dvar_iso_k}. |
---|
| 609 | This results in a term similar to \autoref{eq:perfect-square}, |
---|
[10414] | 610 | \[ |
---|
| 611 | % \label{eq:TScovar} |
---|
| 612 | - \Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} |
---|
| 613 | \left( |
---|
| 614 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 615 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
| 616 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
| 617 | \right) |
---|
| 618 | \left( |
---|
| 619 | \frac{ \delta_{i+ i_p}[S^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
| 620 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
| 621 | \frac{ \delta_{k+k_p} [S^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
| 622 | \right). |
---|
| 623 | \] |
---|
[10354] | 624 | This is symmetrical in $T $ and $S$, so exactly the same term arises from |
---|
| 625 | the discretization of this triad's contribution towards the RHS of \autoref{eq:iso_property3}. |
---|
[2282] | 626 | \end{description} |
---|
[9393] | 627 | |
---|
[10414] | 628 | \subsection{Treatment of the triads at the boundaries} |
---|
| 629 | \label{sec:iso_bdry} |
---|
| 630 | |
---|
[10354] | 631 | The triad slope can only be defined where both the grid boxes centred at the end of the arms exist. |
---|
| 632 | Triads that would poke up through the upper ocean surface into the atmosphere, |
---|
| 633 | or down into the ocean floor, must be masked out. |
---|
| 634 | See \autoref{fig:bdry_triads}. |
---|
| 635 | Surface layer triads $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and $\triad{i+1}{1}{R}{-1/2}{-1/2}$ (blue) that |
---|
| 636 | require density to be specified above the ocean surface are masked (\autoref{fig:bdry_triads}a): |
---|
| 637 | this ensures that lateral tracer gradients produce no flux through the ocean surface. |
---|
| 638 | However, to prevent surface noise, it is customary to retain the $_{11}$ contributions towards |
---|
| 639 | the lateral triad fluxes $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$; |
---|
| 640 | this drives diapycnal tracer fluxes. |
---|
| 641 | Similar comments apply to triads that would intersect the ocean floor (\autoref{fig:bdry_triads}b). |
---|
| 642 | Note that both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and |
---|
| 643 | $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, |
---|
| 644 | i.e.\ the $i,k+1$ $u$-point is masked. |
---|
| 645 | The associated lateral fluxes (grey-black dashed line) are masked if \np{ln\_botmix\_triad}\forcode{ = .false.}, |
---|
| 646 | but left unmasked, giving bottom mixing, if \np{ln\_botmix\_triad}\forcode{ = .true.}. |
---|
[2282] | 647 | |
---|
[10354] | 648 | The default option \np{ln\_botmix\_triad}\forcode{ = .false.} is suitable when the bbl mixing option is enabled |
---|
| 649 | (\key{trabbl}, with \np{nn\_bbl\_ldf}\forcode{ = 1}), or for simple idealized problems. |
---|
| 650 | For setups with topography without bbl mixing, \np{ln\_botmix\_triad}\forcode{ = .true.} may be necessary. |
---|
[3294] | 651 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[10414] | 652 | \begin{figure}[h] |
---|
| 653 | \begin{center} |
---|
[6997] | 654 | \includegraphics[width=0.60\textwidth]{Fig_GRIFF_bdry_triads} |
---|
[10414] | 655 | \caption{ |
---|
| 656 | \protect\label{fig:bdry_triads} |
---|
[10354] | 657 | (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer points (black dots), |
---|
| 658 | and $i+1/2,1$ $u$-point (blue square). |
---|
| 659 | Triad slopes $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and $\triad{i+1}{1}{R}{-1/2}{-1/2}$ (blue) poking through |
---|
| 660 | the ocean surface are masked (faded in figure). |
---|
| 661 | However, the lateral $_{11}$ contributions towards $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and |
---|
| 662 | $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$ (yellow line) are still applied, |
---|
[10414] | 663 | giving diapycnal diffusive fluxes. |
---|
| 664 | \newline |
---|
[3294] | 665 | (b) Both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and |
---|
[10354] | 666 | $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, |
---|
| 667 | i.e.\ the $i,k+1$ $u$-point is masked. |
---|
| 668 | The associated lateral fluxes (grey-black dashed line) are masked if |
---|
| 669 | \protect\np{botmix\_triad}\forcode{ = .false.}, but left unmasked, |
---|
[10414] | 670 | giving bottom mixing, if \protect\np{botmix\_triad}\forcode{ = .true.} |
---|
| 671 | } |
---|
| 672 | \end{center} |
---|
| 673 | \end{figure} |
---|
[3294] | 674 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[9393] | 675 | |
---|
[10414] | 676 | \subsection{ Limiting of the slopes within the interior} |
---|
| 677 | \label{sec:limit} |
---|
| 678 | |
---|
[10354] | 679 | As discussed in \autoref{subsec:LDF_slp_iso}, |
---|
| 680 | iso-neutral slopes relative to geopotentials must be bounded everywhere, |
---|
| 681 | both for consistency with the small-slope approximation and for numerical stability \citep{Cox1987, Griffies_Bk04}. |
---|
| 682 | The bound chosen in \NEMO is applied to each component of the slope separately and |
---|
| 683 | has a value of $1/100$ in the ocean interior. |
---|
[3294] | 684 | %, ramping linearly down above 70~m depth to zero at the surface |
---|
[10354] | 685 | It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to geopotentials |
---|
| 686 | (here the $\sigma_i$ are the slopes of the coordinate surfaces relative to geopotentials) |
---|
| 687 | \autoref{eq:PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate surfaces, so we require |
---|
[10406] | 688 | \[ |
---|
[3294] | 689 | |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01. |
---|
[10406] | 690 | \] |
---|
[3294] | 691 | and then recalculate the slopes $r_i$ relative to coordinates. |
---|
| 692 | Each individual triad slope |
---|
[10354] | 693 | \begin{equation} |
---|
| 694 | \label{eq:Rtilde} |
---|
| 695 | _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}} |
---|
| 696 | \end{equation} |
---|
| 697 | is limited like this and then the corresponding $_i^k\mathbb{R}_{i_p}^{k_p} $ are recalculated and |
---|
| 698 | combined to form the fluxes. |
---|
| 699 | Note that where the slopes have been limited, there is now a non-zero iso-neutral density flux that |
---|
| 700 | drives dianeutral mixing. |
---|
| 701 | In particular this iso-neutral density flux is always downwards, |
---|
| 702 | and so acts to reduce gravitational potential energy. |
---|
[9393] | 703 | |
---|
[10414] | 704 | \subsection{Tapering within the surface mixed layer} |
---|
| 705 | \label{sec:taper} |
---|
| 706 | |
---|
[10354] | 707 | Additional tapering of the iso-neutral fluxes is necessary within the surface mixed layer. |
---|
| 708 | When the Griffies triads are used, we offer two options for this. |
---|
[9393] | 709 | |
---|
[10414] | 710 | \subsubsection{Linear slope tapering within the surface mixed layer} |
---|
| 711 | \label{sec:lintaper} |
---|
| 712 | |
---|
[10354] | 713 | This is the option activated by the default choice \np{ln\_triad\_iso}\forcode{ = .false.}. |
---|
| 714 | Slopes $\tilde{r}_i$ relative to geopotentials are tapered linearly from their value immediately below |
---|
| 715 | the mixed layer to zero at the surface, as described in option (c) of \autoref{fig:eiv_slp}, to values |
---|
[10414] | 716 | \begin{equation} |
---|
| 717 | \label{eq:rmtilde} |
---|
| 718 | \rMLt = -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for } z>-h, |
---|
| 719 | \end{equation} |
---|
| 720 | and then the $r_i$ relative to vertical coordinate surfaces are appropriately adjusted to |
---|
| 721 | \[ |
---|
| 722 | % \label{eq:rm} |
---|
| 723 | \rML =\rMLt -\sigma_i \quad \text{ for } z>-h. |
---|
| 724 | \] |
---|
[3294] | 725 | Thus the diffusion operator within the mixed layer is given by: |
---|
[10414] | 726 | \[ |
---|
| 727 | % \label{eq:iso_tensor_ML} |
---|
| 728 | D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad |
---|
| 729 | \mbox{with}\quad \;\;\Re =\left( {{ |
---|
| 730 | \begin{array}{*{20}c} |
---|
| 731 | 1 \hfill & 0 \hfill & {-\rML[1]}\hfill \\ |
---|
| 732 | 0 \hfill & 1 \hfill & {-\rML[2]} \hfill \\ |
---|
| 733 | {-\rML[1]}\hfill & {-\rML[2]} \hfill & {\rML[1]^2+\rML[2]^2} \hfill |
---|
| 734 | \end{array} |
---|
| 735 | }} \right) |
---|
| 736 | \] |
---|
[3294] | 737 | |
---|
[10354] | 738 | This slope tapering gives a natural connection between tracer in the mixed-layer and |
---|
| 739 | in isopycnal layers immediately below, in the thermocline. |
---|
| 740 | It is consistent with the way the $\tilde{r}_i$ are tapered within the mixed layer |
---|
| 741 | (see \autoref{sec:taperskew} below) so as to ensure a uniform GM eddy-induced velocity throughout the mixed layer. |
---|
| 742 | However, it gives a downwards density flux and so acts so as to reduce potential energy in the same way as |
---|
| 743 | does the slope limiting discussed above in \autoref{sec:limit}. |
---|
[3294] | 744 | |
---|
[10354] | 745 | As in \autoref{sec:limit} above, the tapering \autoref{eq:rmtilde} is applied separately to |
---|
| 746 | each triad $_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the $_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted. |
---|
| 747 | For clarity, we assume $z$-coordinates in the following; |
---|
| 748 | the conversion from $\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as |
---|
| 749 | described above by \autoref{eq:Rtilde}. |
---|
[3294] | 750 | \begin{enumerate} |
---|
[10354] | 751 | \item |
---|
| 752 | Mixed-layer depth is defined so as to avoid including regions of weak vertical stratification in |
---|
| 753 | the slope definition. |
---|
| 754 | At each $i,j$ (simplified to $i$ in \autoref{fig:MLB_triad}), |
---|
| 755 | we define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer, |
---|
| 756 | $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that |
---|
| 757 | the potential density ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, |
---|
| 758 | where $i,k_{10}$ is the tracer gridbox within which the depth reaches 10~m. |
---|
| 759 | See the left side of \autoref{fig:MLB_triad}. |
---|
| 760 | We use the $k_{10}$-gridbox instead of the surface gridbox to avoid problems e.g.\ with thin daytime mixed-layers. |
---|
| 761 | Currently we use the same $\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is used to |
---|
| 762 | output the diagnosed mixed-layer depth $h_{\mathrm{ML}}=|z_{W}|_{k_{\mathrm{ML}}+1/2}$, |
---|
| 763 | the depth of the $w$-point above the $i,k_{\mathrm{ML}}$ tracer point. |
---|
| 764 | \item |
---|
| 765 | We define `basal' triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ as |
---|
| 766 | the slopes of those triads whose vertical `arms' go down from the $i,k_{\mathrm{ML}}$ tracer point to |
---|
| 767 | the $i,k_{\mathrm{ML}}-1$ tracer point below. |
---|
| 768 | This is to ensure that the vertical density gradients associated with |
---|
| 769 | these basal triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are representative of the thermocline. |
---|
| 770 | The four basal triads defined in the bottom part of \autoref{fig:MLB_triad} are then |
---|
[10414] | 771 | \begin{align*} |
---|
| 772 | {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &= |
---|
| 773 | {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}, |
---|
| 774 | % \label{eq:Rbase} |
---|
| 775 | \\ |
---|
| 776 | \intertext{with e.g.\ the green triad} |
---|
| 777 | {\:}_i{\mathbb{R}_{\mathrm{base}}}_{1/2}^{-1/2}&= |
---|
| 778 | {\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}. |
---|
| 779 | \end{align*} |
---|
[10354] | 780 | The vertical flux associated with each of these triads passes through |
---|
| 781 | the $w$-point $i,k_{\mathrm{ML}}-1/2$ lying \emph{below} the $i,k_{\mathrm{ML}}$ tracer point, so it is this depth |
---|
[10414] | 782 | \[ |
---|
| 783 | % \label{eq:zbase} |
---|
[3294] | 784 | {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2} |
---|
[10414] | 785 | \] |
---|
[10354] | 786 | one gridbox deeper than the diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper the slopes in |
---|
| 787 | \autoref{eq:rmtilde}. |
---|
| 788 | \item |
---|
| 789 | Finally, we calculate the adjusted triads ${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within |
---|
| 790 | the mixed layer, by multiplying the appropriate ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ by |
---|
| 791 | the ratio of the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_{\mathrm{base}}}_{\,i}$. |
---|
| 792 | For instance the green triad centred on $i,k$ |
---|
[10414] | 793 | \begin{align*} |
---|
| 794 | {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,1/2}^{-1/2} &= |
---|
| 795 | \frac{{z_w}_{k-1/2}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2} \\ |
---|
| 796 | \intertext{and more generally} |
---|
| 797 | {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p} &= |
---|
| 798 | \frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}. |
---|
| 799 | % \label{eq:RML} |
---|
| 800 | \end{align*} |
---|
[3294] | 801 | \end{enumerate} |
---|
| 802 | |
---|
| 803 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 804 | \begin{figure}[h] |
---|
[9364] | 805 | % \fcapside { |
---|
[10414] | 806 | \caption{ |
---|
| 807 | \protect\label{fig:MLB_triad} |
---|
[10354] | 808 | Definition of mixed-layer depth and calculation of linearly tapered triads. |
---|
| 809 | The figure shows a water column at a given $i,j$ (simplified to $i$), with the ocean surface at the top. |
---|
| 810 | Tracer points are denoted by bullets, and black lines the edges of the tracer cells; |
---|
[10414] | 811 | $k$ increases upwards. |
---|
| 812 | \newline |
---|
[10354] | 813 | \hspace{5 em} |
---|
| 814 | We define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer, |
---|
| 815 | $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that |
---|
| 816 | ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, |
---|
| 817 | where $i,k_{10}$ is the tracer gridbox within which the depth reaches 10~m. |
---|
| 818 | We calculate the triad slopes within the mixed layer by linearly tapering them from zero |
---|
| 819 | (at the surface) to the `basal' slopes, |
---|
| 820 | the slopes of the four triads passing through the $w$-point $i,k_{\mathrm{ML}}-1/2$ (blue square), |
---|
| 821 | ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$. |
---|
| 822 | Triads with different $i_p,k_p$, denoted by different colours, |
---|
| 823 | (e.g. the green triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.} |
---|
[10414] | 824 | % } |
---|
| 825 | \includegraphics[width=0.60\textwidth]{Fig_GRIFF_MLB_triads} |
---|
[3294] | 826 | \end{figure} |
---|
| 827 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 828 | |
---|
[9393] | 829 | \subsubsection{Additional truncation of skew iso-neutral flux components} |
---|
[9407] | 830 | \label{subsec:Gerdes-taper} |
---|
[10414] | 831 | |
---|
[10354] | 832 | The alternative option is activated by setting \np{ln\_triad\_iso} = true. |
---|
| 833 | This retains the same tapered slope $\rML$ described above for the calculation of the $_{33}$ term of |
---|
| 834 | the iso-neutral diffusion tensor (the vertical tracer flux driven by vertical tracer gradients), |
---|
| 835 | but replaces the $\rML$ in the skew term by |
---|
[3294] | 836 | \begin{equation} |
---|
[9407] | 837 | \label{eq:rm*} |
---|
[3294] | 838 | \rML^*=\left.\rMLt^2\right/\tilde{r}_i-\sigma_i, |
---|
| 839 | \end{equation} |
---|
| 840 | giving a ML diffusive operator |
---|
[10414] | 841 | \[ |
---|
| 842 | % \label{eq:iso_tensor_ML2} |
---|
| 843 | D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad |
---|
| 844 | \mbox{with}\quad \;\;\Re =\left( {{ |
---|
| 845 | \begin{array}{*{20}c} |
---|
| 846 | 1 \hfill & 0 \hfill & {-\rML[1]^*}\hfill \\ |
---|
| 847 | 0 \hfill & 1 \hfill & {-\rML[2]^*} \hfill \\ |
---|
| 848 | {-\rML[1]^*}\hfill & {-\rML[2]^*} \hfill & {\rML[1]^2+\rML[2]^2} \hfill \\ |
---|
| 849 | \end{array} |
---|
| 850 | }} \right). |
---|
| 851 | \] |
---|
[3294] | 852 | This operator |
---|
[10354] | 853 | \footnote{ |
---|
| 854 | To ensure good behaviour where horizontal density gradients are weak, |
---|
| 855 | we in fact follow \citet{Gerdes1991} and |
---|
[10414] | 856 | set $\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$. |
---|
| 857 | } |
---|
[10354] | 858 | then has the property it gives no vertical density flux, and so does not change the potential energy. |
---|
| 859 | This approach is similar to multiplying the iso-neutral diffusion coefficient by |
---|
| 860 | $\tilde{r}_{\mathrm{max}}^{-2}\tilde{r}_i^{-2}$ for steep slopes, |
---|
| 861 | as suggested by \citet{Gerdes1991} (see also \citet{Griffies_Bk04}). |
---|
[3294] | 862 | Again it is applied separately to each triad $_i^k\mathbb{R}_{i_p}^{k_p}$ |
---|
| 863 | |
---|
[10354] | 864 | In practice, this approach gives weak vertical tracer fluxes through the mixed-layer, |
---|
| 865 | as well as vanishing density fluxes. |
---|
| 866 | While it is theoretically advantageous that it does not change the potential energy, |
---|
| 867 | it may give a discontinuity between the fluxes within the mixed-layer (purely horizontal) and |
---|
| 868 | just below (along iso-neutral surfaces). |
---|
[3294] | 869 | % This may give strange looking results, |
---|
| 870 | % particularly where the mixed-layer depth varies strongly laterally. |
---|
[2282] | 871 | % ================================================================ |
---|
| 872 | % Skew flux formulation for Eddy Induced Velocity : |
---|
| 873 | % ================================================================ |
---|
[10414] | 874 | \section{Eddy induced advection formulated as a skew flux} |
---|
| 875 | \label{sec:skew-flux} |
---|
[2282] | 876 | |
---|
[10414] | 877 | \subsection{Continuous skew flux formulation} |
---|
| 878 | \label{sec:continuous-skew-flux} |
---|
[3294] | 879 | |
---|
[10354] | 880 | When Gent and McWilliams's [1990] diffusion is used, an additional advection term is added. |
---|
| 881 | The associated velocity is the so called eddy induced velocity, |
---|
| 882 | the formulation of which depends on the slopes of iso-neutral surfaces. |
---|
| 883 | Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, |
---|
| 884 | $i.e.$ \autoref{eq:ldfslp_geo} is used in $z$-coordinate, |
---|
| 885 | and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates. |
---|
[2282] | 886 | |
---|
[3294] | 887 | The eddy induced velocity is given by: |
---|
[10414] | 888 | \begin{subequations} |
---|
| 889 | % \label{eq:eiv} |
---|
| 890 | \begin{equation} |
---|
| 891 | \label{eq:eiv_v} |
---|
| 892 | \begin{split} |
---|
| 893 | u^* & = - \frac{1}{e_{3}}\; \partial_i\psi_1, \\ |
---|
| 894 | v^* & = - \frac{1}{e_{3}}\; \partial_j\psi_2, \\ |
---|
| 895 | w^* & = \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i \left( e_{2} \, \psi_1\right) |
---|
| 896 | + \partial_j \left( e_{1} \, \psi_2\right) \right\}, |
---|
| 897 | \end{split} |
---|
| 898 | \end{equation} |
---|
| 899 | where the streamfunctions $\psi_i$ are given by |
---|
| 900 | \begin{equation} |
---|
| 901 | \label{eq:eiv_psi} |
---|
| 902 | \begin{split} |
---|
| 903 | \psi_1 & = A_{e} \; \tilde{r}_1, \\ |
---|
| 904 | \psi_2 & = A_{e} \; \tilde{r}_2, |
---|
| 905 | \end{split} |
---|
| 906 | \end{equation} |
---|
[3294] | 907 | \end{subequations} |
---|
[10354] | 908 | with $A_{e}$ the eddy induced velocity coefficient, |
---|
| 909 | and $\tilde{r}_1$ and $\tilde{r}_2$ the slopes between the iso-neutral and the geopotential surfaces. |
---|
[2282] | 910 | |
---|
[10354] | 911 | The traditional way to implement this additional advection is to add it to the Eulerian velocity prior to |
---|
| 912 | computing the tracer advection. |
---|
| 913 | This is implemented if \key{traldf\_eiv} is set in the default implementation, |
---|
| 914 | where \np{ln\_traldf\_triad} is set false. |
---|
| 915 | This allows us to take advantage of all the advection schemes offered for the tracers |
---|
| 916 | (see \autoref{sec:TRA_adv}) and not just a $2^{nd}$ order advection scheme. |
---|
| 917 | This is particularly useful for passive tracers where |
---|
| 918 | \emph{positivity} of the advection scheme is of paramount importance. |
---|
[2282] | 919 | |
---|
[10354] | 920 | However, when \np{ln\_traldf\_triad} is set true, |
---|
| 921 | \NEMO instead implements eddy induced advection according to the so-called skew form \citep{Griffies_JPO98}. |
---|
| 922 | It is based on a transformation of the advective fluxes using the non-divergent nature of the eddy induced velocity. |
---|
| 923 | For example in the (\textbf{i},\textbf{k}) plane, |
---|
| 924 | the tracer advective fluxes per unit area in $ijk$ space can be transformed as follows: |
---|
[2282] | 925 | \begin{flalign*} |
---|
[10414] | 926 | \begin{split} |
---|
| 927 | \textbf{F}_{\mathrm{eiv}}^T = |
---|
| 928 | \begin{pmatrix} |
---|
| 929 | {e_{2}\,e_{3}\; u^*} \\ |
---|
| 930 | {e_{1}\,e_{2}\; w^*} |
---|
| 931 | \end{pmatrix} \; T |
---|
| 932 | &= |
---|
| 933 | \begin{pmatrix} |
---|
| 934 | { - \partial_k \left( e_{2} \,\psi_1 \right) \; T \;} \\ |
---|
| 935 | {+ \partial_i \left( e_{2} \, \psi_1 \right) \; T \;} |
---|
| 936 | \end{pmatrix} \\ |
---|
| 937 | &= |
---|
| 938 | \begin{pmatrix} |
---|
| 939 | { - \partial_k \left( e_{2} \, \psi_1 \; T \right) \;} \\ |
---|
| 940 | {+ \partial_i \left( e_{2} \,\psi_1 \; T \right) \;} |
---|
| 941 | \end{pmatrix} |
---|
| 942 | + |
---|
| 943 | \begin{pmatrix} |
---|
| 944 | {+ e_{2} \, \psi_1 \; \partial_k T} \\ |
---|
| 945 | { - e_{2} \, \psi_1 \; \partial_i T} |
---|
| 946 | \end{pmatrix} |
---|
| 947 | \end{split} |
---|
[2282] | 948 | \end{flalign*} |
---|
[10354] | 949 | and since the eddy induced velocity field is non-divergent, |
---|
| 950 | we end up with the skew form of the eddy induced advective fluxes per unit area in $ijk$ space: |
---|
[10414] | 951 | \begin{equation} |
---|
| 952 | \label{eq:eiv_skew_ijk} |
---|
| 953 | \textbf{F}_\mathrm{eiv}^T = |
---|
| 954 | \begin{pmatrix} |
---|
| 955 | {+ e_{2} \, \psi_1 \; \partial_k T} \\ |
---|
| 956 | { - e_{2} \, \psi_1 \; \partial_i T} |
---|
| 957 | \end{pmatrix} |
---|
[2282] | 958 | \end{equation} |
---|
[3294] | 959 | The total fluxes per unit physical area are then |
---|
[10414] | 960 | \begin{equation} |
---|
| 961 | \label{eq:eiv_skew_physical} |
---|
| 962 | \begin{split} |
---|
| 963 | f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T \\ |
---|
| 964 | f^*_2 & = \frac{1}{e_{3}}\; \psi_2 \partial_k T \\ |
---|
| 965 | f^*_3 & = -\frac{1}{e_{1}e_{2}}\; \left\{ e_{2} \psi_1 \partial_i T + e_{1} \psi_2 \partial_j T \right\}. |
---|
[3294] | 966 | \end{split} |
---|
| 967 | \end{equation} |
---|
[10354] | 968 | Note that \autoref{eq:eiv_skew_physical} takes the same form whatever the vertical coordinate, |
---|
| 969 | though of course the slopes $\tilde{r}_i$ which define the $\psi_i$ in \autoref{eq:eiv_psi} are relative to |
---|
| 970 | geopotentials. |
---|
| 971 | The tendency associated with eddy induced velocity is then simply the convergence of the fluxes |
---|
| 972 | (\autoref{eq:eiv_skew_ijk}, \autoref{eq:eiv_skew_physical}), so |
---|
[10414] | 973 | \[ |
---|
| 974 | % \label{eq:skew_eiv_conv} |
---|
| 975 | \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 } \left[ |
---|
| 976 | \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right) |
---|
| 977 | + \frac{\partial}{\partial j} \left( e_1 \; |
---|
| 978 | \psi_2 \partial_k T\right) |
---|
| 979 | - \frac{\partial}{\partial k} \left( e_{2} \psi_1 \partial_i T |
---|
| 980 | + e_{1} \psi_2 \partial_j T \right) \right] |
---|
| 981 | \] |
---|
[10354] | 982 | It naturally conserves the tracer content, as it is expressed in flux form. |
---|
| 983 | Since it has the same divergence as the advective form it also preserves the tracer variance. |
---|
[2282] | 984 | |
---|
[9393] | 985 | \subsection{Discrete skew flux formulation} |
---|
[10414] | 986 | |
---|
[10354] | 987 | The skew fluxes in (\autoref{eq:eiv_skew_physical}, \autoref{eq:eiv_skew_ijk}), |
---|
| 988 | like the off-diagonal terms (\autoref{eq:i13c}, \autoref{eq:i31c}) of the small angle diffusion tensor, |
---|
| 989 | are best expressed in terms of the triad slopes, as in \autoref{fig:ISO_triad} and |
---|
| 990 | (\autoref{eq:i13}, \autoref{eq:i31}); |
---|
| 991 | but now in terms of the triad slopes $\tilde{\mathbb{R}}$ relative to geopotentials instead of |
---|
| 992 | the $\mathbb{R}$ relative to coordinate surfaces. |
---|
| 993 | The discrete form of \autoref{eq:eiv_skew_ijk} using the slopes \autoref{eq:R} and |
---|
[3294] | 994 | defining $A_e$ at $T$-points is then given by: |
---|
[2282] | 995 | |
---|
[10414] | 996 | \begin{subequations} |
---|
| 997 | % \label{eq:allskewflux} |
---|
| 998 | \begin{flalign*} |
---|
| 999 | % \label{eq:vect_skew_flux} |
---|
| 1000 | \vect{F}_{\mathrm{eiv}}(T) &\equiv \sum_{\substack{i_p,\,k_p}} |
---|
[3294] | 1001 | \begin{pmatrix} |
---|
[10414] | 1002 | {_{i+1/2-i_p}^k {\mathbb{S}_u}_{i_p}^{k_p} } (T) \\ \\ |
---|
[3294] | 1003 | {_i^{k+1/2-k_p} {\mathbb{S}_w}_{i_p}^{k_p} } (T) \\ |
---|
| 1004 | \end{pmatrix}, |
---|
[10414] | 1005 | \end{flalign*} |
---|
[10354] | 1006 | where the skew flux in the $i$-direction associated with a given triad is (\autoref{eq:latflux-triad}, |
---|
| 1007 | \autoref{eq:triadfluxu}): |
---|
[3294] | 1008 | \begin{align} |
---|
[9407] | 1009 | \label{eq:skewfluxu} |
---|
[3294] | 1010 | _i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) &= + \quarter {A_e}_i^k{ |
---|
[10414] | 1011 | \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
| 1012 | \ {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} \ |
---|
| 1013 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }, \\ |
---|
| 1014 | \intertext{ |
---|
| 1015 | and \autoref{eq:triadfluxw} in the $k$-direction, changing the sign |
---|
| 1016 | to be consistent with \autoref{eq:eiv_skew_ijk}: |
---|
| 1017 | } |
---|
[3294] | 1018 | _i^k {\mathbb{S}_w}_{i_p}^{k_p} (T) |
---|
[10414] | 1019 | &= -\quarter {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}} |
---|
| 1020 | {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:skewfluxw} |
---|
[3294] | 1021 | \end{align} |
---|
| 1022 | \end{subequations} |
---|
[2282] | 1023 | |
---|
[10354] | 1024 | Such a discretisation is consistent with the iso-neutral operator as it uses the same definition for the slopes. |
---|
| 1025 | It also ensures the following two key properties. |
---|
[9393] | 1026 | |
---|
[3294] | 1027 | \subsubsection{No change in tracer variance} |
---|
[10414] | 1028 | |
---|
[10354] | 1029 | The discretization conserves tracer variance, $i.e.$ it does not include a diffusive component but is a `pure' advection term. |
---|
| 1030 | This can be seen %either from Appendix \autoref{apdx:eiv_skew} or |
---|
| 1031 | by considering the fluxes associated with a given triad slope $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. |
---|
| 1032 | For, following \autoref{subsec:variance} and \autoref{eq:dvar_iso_i}, |
---|
| 1033 | the associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$ drives a net rate of change of variance, |
---|
| 1034 | summed over the two $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of |
---|
[3294] | 1035 | \begin{equation} |
---|
[10414] | 1036 | \label{eq:dvar_eiv_i} |
---|
[3294] | 1037 | _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], |
---|
| 1038 | \end{equation} |
---|
[10354] | 1039 | while the associated vertical skew-flux gives a variance change summed over |
---|
| 1040 | the $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of |
---|
[3294] | 1041 | \begin{equation} |
---|
[10414] | 1042 | \label{eq:dvar_eiv_k} |
---|
[3294] | 1043 | _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. |
---|
| 1044 | \end{equation} |
---|
[10354] | 1045 | Inspection of the definitions (\autoref{eq:skewfluxu}, \autoref{eq:skewfluxw}) shows that |
---|
| 1046 | these two variance changes (\autoref{eq:dvar_eiv_i}, \autoref{eq:dvar_eiv_k}) sum to zero. |
---|
| 1047 | Hence the two fluxes associated with each triad make no net contribution to the variance budget. |
---|
[2282] | 1048 | |
---|
[3294] | 1049 | \subsubsection{Reduction in gravitational PE} |
---|
[10414] | 1050 | |
---|
[10354] | 1051 | The vertical density flux associated with the vertical skew-flux always has the same sign as |
---|
| 1052 | the vertical density gradient; |
---|
| 1053 | thus, so long as the fluid is stable (the vertical density gradient is negative) |
---|
| 1054 | the vertical density flux is negative (downward) and hence reduces the gravitational PE. |
---|
[2282] | 1055 | |
---|
[3294] | 1056 | For the change in gravitational PE driven by the $k$-flux is |
---|
| 1057 | \begin{align} |
---|
[9407] | 1058 | \label{eq:vert_densityPE} |
---|
[3294] | 1059 | g {e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) |
---|
| 1060 | &=g {e_{3w}}_{\,i}^{\,k+k_p}\left[-\alpha _i^k {\:}_i^k |
---|
| 1061 | {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k |
---|
| 1062 | {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\ |
---|
[10414] | 1063 | \intertext{Substituting ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from \autoref{eq:skewfluxw}, gives} |
---|
| 1064 | % and separating out |
---|
| 1065 | % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$, |
---|
| 1066 | % gives two terms. The |
---|
| 1067 | % first $\rtriad{R}$ term (the only term for $z$-coordinates) is: |
---|
| 1068 | &=-\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} |
---|
| 1069 | \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 \\ |
---|
| 1070 | &=+\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
| 1071 | \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}} |
---|
| 1072 | \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}}, |
---|
[3294] | 1073 | \end{align} |
---|
[10354] | 1074 | using the definition of the triad slope $\rtriad{R}$, \autoref{eq:R} to |
---|
| 1075 | express $-\alpha _i^k\delta_{i+ i_p}[T^k]+\beta_i^k\delta_{i+ i_p}[S^k]$ in terms of |
---|
| 1076 | $-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$. |
---|
[2282] | 1077 | |
---|
[3294] | 1078 | Where the coordinates slope, the $i$-flux gives a PE change |
---|
| 1079 | \begin{multline} |
---|
[9407] | 1080 | \label{eq:lat_densityPE} |
---|
[10414] | 1081 | g \delta_{i+i_p}[z_T^k] |
---|
| 1082 | \left[ |
---|
| 1083 | -\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) |
---|
| 1084 | \right] \\ |
---|
| 1085 | = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
| 1086 | \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
| 1087 | \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) |
---|
| 1088 | \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}}, |
---|
[3294] | 1089 | \end{multline} |
---|
[10354] | 1090 | (using \autoref{eq:skewfluxu}) and so the total PE change \autoref{eq:vert_densityPE} + |
---|
| 1091 | \autoref{eq:lat_densityPE} associated with the triad fluxes is |
---|
[10414] | 1092 | \begin{multline*} |
---|
| 1093 | % \label{eq:tot_densityPE} |
---|
[3294] | 1094 | g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) + |
---|
[10414] | 1095 | g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\ |
---|
| 1096 | = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
| 1097 | \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 |
---|
| 1098 | \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}}. |
---|
| 1099 | \end{multline*} |
---|
[3294] | 1100 | Where the fluid is stable, with $-\alpha_i^k \delta_{k+ k_p}[T^i]+ |
---|
| 1101 | \beta_i^k\delta_{k+ k_p}[S^i]<0$, this PE change is negative. |
---|
[2282] | 1102 | |
---|
[10414] | 1103 | \subsection{Treatment of the triads at the boundaries} |
---|
| 1104 | \label{sec:skew_bdry} |
---|
| 1105 | |
---|
[10354] | 1106 | Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes are masked at the boundaries |
---|
| 1107 | in exactly the same way as are the triad slopes \rtriad{R} used for the iso-neutral diffusive fluxes, |
---|
| 1108 | as described in \autoref{sec:iso_bdry} and \autoref{fig:bdry_triads}. |
---|
| 1109 | Thus surface layer triads $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are masked, |
---|
| 1110 | and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when |
---|
| 1111 | either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point is masked. |
---|
| 1112 | The namelist parameter \np{ln\_botmix\_triad} has no effect on the eddy-induced skew-fluxes. |
---|
[2282] | 1113 | |
---|
[10414] | 1114 | \subsection{Limiting of the slopes within the interior} |
---|
| 1115 | \label{sec:limitskew} |
---|
| 1116 | |
---|
[10354] | 1117 | Presently, the iso-neutral slopes $\tilde{r}_i$ relative to geopotentials are limited to be less than $1/100$, |
---|
| 1118 | exactly as in calculating the iso-neutral diffusion, \S \autoref{sec:limit}. |
---|
| 1119 | Each individual triad \rtriadt{R} is so limited. |
---|
[2282] | 1120 | |
---|
[10414] | 1121 | \subsection{Tapering within the surface mixed layer} |
---|
| 1122 | \label{sec:taperskew} |
---|
| 1123 | |
---|
[10354] | 1124 | The slopes $\tilde{r}_i$ relative to geopotentials (and thus the individual triads \rtriadt{R}) |
---|
| 1125 | are always tapered linearly from their value immediately below the mixed layer to zero at the surface |
---|
| 1126 | \autoref{eq:rmtilde}, as described in \autoref{sec:lintaper}. |
---|
| 1127 | This is option (c) of \autoref{fig:eiv_slp}. |
---|
| 1128 | This linear tapering for the slopes used to calculate the eddy-induced fluxes is unaffected by |
---|
| 1129 | the value of \np{ln\_triad\_iso}. |
---|
[2282] | 1130 | |
---|
[10354] | 1131 | The justification for this linear slope tapering is that, for $A_e$ that is constant or varies only in |
---|
| 1132 | the horizontal (the most commonly used options in \NEMO: see \autoref{sec:LDF_coef}), |
---|
| 1133 | it is equivalent to a horizontal eiv (eddy-induced velocity) that is uniform within the mixed layer |
---|
| 1134 | \autoref{eq:eiv_v}. |
---|
| 1135 | This ensures that the eiv velocities do not restratify the mixed layer \citep{Treguier1997,Danabasoglu_al_2008}. |
---|
| 1136 | Equivantly, in terms of the skew-flux formulation we use here, |
---|
| 1137 | the linear slope tapering within the mixed-layer gives a linearly varying vertical flux, |
---|
| 1138 | and so a tracer convergence uniform in depth |
---|
| 1139 | (the horizontal flux convergence is relatively insignificant within the mixed-layer). |
---|
[3294] | 1140 | |
---|
[10414] | 1141 | \subsection{Streamfunction diagnostics} |
---|
| 1142 | \label{sec:sfdiag} |
---|
| 1143 | |
---|
[10354] | 1144 | Where the namelist parameter \np{ln\_traldf\_gdia}\forcode{ = .true.}, |
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| 1145 | diagnosed mean eddy-induced velocities are output. |
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| 1146 | Each time step, streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at |
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| 1147 | $uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$ (integer $i$, integer +1/2 $j$, integer +1/2 $k$) |
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| 1148 | points (see Table \autoref{tab:cell}) respectively. |
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| 1149 | We follow \citep{Griffies_Bk04} and calculate the streamfunction at a given $uw$-point from |
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| 1150 | the surrounding four triads according to: |
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[10414] | 1151 | \[ |
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| 1152 | % \label{eq:sfdiagi} |
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[3294] | 1153 | {\psi_1}_{i+1/2}^{k+1/2}={\quarter}\sum_{\substack{i_p,\,k_p}} |
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| 1154 | {A_e}_{i+1/2-i_p}^{k+1/2-k_p}\:\triadd{i+1/2-i_p}{k+1/2-k_p}{R}{i_p}{k_p}. |
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[10414] | 1155 | \] |
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[3294] | 1156 | The streamfunction $\psi_1$ is calculated similarly at $vw$ points. |
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[10354] | 1157 | The eddy-induced velocities are then calculated from the straightforward discretisation of \autoref{eq:eiv_v}: |
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[10414] | 1158 | \[ |
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| 1159 | % \label{eq:eiv_v_discrete} |
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| 1160 | \begin{split} |
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| 1161 | {u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k}}\left({\psi_1}_{i+1/2}^{k+1/2}-{\psi_1}_{i+1/2}^{k+1/2}\right), \\ |
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| 1162 | {v^*}_{j+1/2}^{k} & = - \frac{1}{{e_{3v}}_{j}^{k}}\left({\psi_2}_{j+1/2}^{k+1/2}-{\psi_2}_{j+1/2}^{k+1/2}\right), \\ |
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| 1163 | {w^*}_{i,j}^{k+1/2} & = \frac{1}{e_{1t}e_{2t}}\; \left\{ |
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| 1164 | {e_{2u}}_{i+1/2}^{k+1/2} \,{\psi_1}_{i+1/2}^{k+1/2} - |
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| 1165 | {e_{2u}}_{i-1/2}^{k+1/2} \,{\psi_1}_{i-1/2}^{k+1/2} \right. + \\ |
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| 1166 | \phantom{=} & \qquad\qquad\left. {e_{2v}}_{j+1/2}^{k+1/2} \,{\psi_2}_{j+1/2}^{k+1/2} - {e_{2v}}_{j-1/2}^{k+1/2} \,{\psi_2}_{j-1/2}^{k+1/2} \right\}, |
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| 1167 | \end{split} |
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| 1168 | \] |
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| 1169 | |
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| 1170 | \biblio |
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| 1171 | |
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[6997] | 1172 | \end{document} |
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