[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
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| 2 | |
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[6997] | 3 | \begin{document} |
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[707] | 4 | |
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| 5 | % ================================================================ |
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[10414] | 6 | % Chapter Lateral Ocean Physics (LDF) |
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[707] | 7 | % ================================================================ |
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| 8 | \chapter{Lateral Ocean Physics (LDF)} |
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[9407] | 9 | \label{chap:LDF} |
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[10414] | 10 | |
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[11435] | 11 | \chaptertoc |
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[707] | 12 | |
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[2282] | 13 | \newpage |
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[707] | 14 | |
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[11543] | 15 | The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:MB_zdf} and |
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[10354] | 16 | their discrete formulation in \autoref{sec:TRA_ldf} and \autoref{sec:DYN_ldf}). |
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| 17 | In this section we further discuss each lateral physics option. |
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| 18 | Choosing one lateral physics scheme means for the user defining, |
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| 19 | (1) the type of operator used (laplacian or bilaplacian operators, or no lateral mixing term); |
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| 20 | (2) the direction along which the lateral diffusive fluxes are evaluated |
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| 21 | (model level, geopotential or isopycnal surfaces); and |
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| 22 | (3) the space and time variations of the eddy coefficients. |
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| 23 | These three aspects of the lateral diffusion are set through namelist parameters |
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[11435] | 24 | (see the \nam{tra\_ldf} and \nam{dyn\_ldf} below). |
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[11558] | 25 | Note that this chapter describes the standard implementation of iso-neutral tracer mixing. |
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[11537] | 26 | Griffies's implementation, which is used if \np{ln\_traldf\_triad}\forcode{=.true.}, |
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[11543] | 27 | is described in \autoref{apdx:TRIADS} |
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[999] | 28 | |
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[11336] | 29 | %-----------------------------------namtra_ldf - namdyn_ldf-------------------------------------------- |
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[10146] | 30 | |
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[707] | 31 | %-------------------------------------------------------------------------------------------------------------- |
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| 32 | |
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[11336] | 33 | % ================================================================ |
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| 34 | % Lateral Mixing Operator |
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| 35 | % ================================================================ |
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[11565] | 36 | \section[Lateral mixing operators]{Lateral mixing operators} |
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[11336] | 37 | \label{sec:LDF_op} |
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| 38 | We remind here the different lateral mixing operators that can be used. Further details can be found in \autoref{subsec:TRA_ldf_op} and \autoref{sec:DYN_ldf}. |
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[707] | 39 | |
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[11565] | 40 | \subsection[No lateral mixing (\forcode{ln_traldf_OFF} \& \forcode{ln_dynldf_OFF})]{No lateral mixing (\protect\np{ln\_traldf\_OFF} \& \protect\np{ln\_dynldf\_OFF})} |
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[11336] | 41 | |
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[11558] | 42 | It is possible to run without explicit lateral diffusion on tracers (\protect\np{ln\_traldf\_OFF}\forcode{=.true.}) and/or |
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| 43 | momentum (\protect\np{ln\_dynldf\_OFF}\forcode{=.true.}). The latter option is even recommended if using the |
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[11537] | 44 | UBS advection scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{=.true.}, |
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[11336] | 45 | see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes. |
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| 46 | |
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[11565] | 47 | \subsection[Laplacian mixing (\forcode{ln_traldf_lap} \& \forcode{ln_dynldf_lap})]{Laplacian mixing (\protect\np{ln\_traldf\_lap} \& \protect\np{ln\_dynldf\_lap})} |
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[11558] | 48 | Setting \protect\np{ln\_traldf\_lap}\forcode{=.true.} and/or \protect\np{ln\_dynldf\_lap}\forcode{=.true.} enables |
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| 49 | a second order diffusion on tracers and momentum respectively. Note that in \NEMO\ 4, one can not combine |
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[11336] | 50 | Laplacian and Bilaplacian operators for the same variable. |
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| 51 | |
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[11565] | 52 | \subsection[Bilaplacian mixing (\forcode{ln_traldf_blp} \& \forcode{ln_dynldf_blp})]{Bilaplacian mixing (\protect\np{ln\_traldf\_blp} \& \protect\np{ln\_dynldf\_blp})} |
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[11558] | 53 | Setting \protect\np{ln\_traldf\_blp}\forcode{=.true.} and/or \protect\np{ln\_dynldf\_blp}\forcode{=.true.} enables |
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| 54 | a fourth order diffusion on tracers and momentum respectively. It is implemented by calling the above Laplacian operator twice. |
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[11435] | 55 | We stress again that from \NEMO\ 4, the simultaneous use Laplacian and Bilaplacian operators is not allowed. |
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[11336] | 56 | |
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[707] | 57 | % ================================================================ |
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| 58 | % Direction of lateral Mixing |
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| 59 | % ================================================================ |
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[11565] | 60 | \section[Direction of lateral mixing (\textit{ldfslp.F90})]{Direction of lateral mixing (\protect\mdl{ldfslp})} |
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[9407] | 61 | \label{sec:LDF_slp} |
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[707] | 62 | |
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[817] | 63 | %%% |
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[10414] | 64 | \gmcomment{ |
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| 65 | we should emphasize here that the implementation is a rather old one. |
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[11123] | 66 | Better work can be achieved by using \citet{griffies.gnanadesikan.ea_JPO98, griffies_bk04} iso-neutral scheme. |
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[10414] | 67 | } |
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[707] | 68 | |
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[10354] | 69 | A direction for lateral mixing has to be defined when the desired operator does not act along the model levels. |
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| 70 | This occurs when $(a)$ horizontal mixing is required on tracer or momentum |
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| 71 | (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) in $s$- or mixed $s$-$z$- coordinates, |
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| 72 | and $(b)$ isoneutral mixing is required whatever the vertical coordinate is. |
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| 73 | This direction of mixing is defined by its slopes in the \textbf{i}- and \textbf{j}-directions at the face of |
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| 74 | the cell of the quantity to be diffused. |
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| 75 | For a tracer, this leads to the following four slopes: |
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[11543] | 76 | $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:TRA_ldf_iso}), |
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[10354] | 77 | while for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and |
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[11558] | 78 | $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. |
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[707] | 79 | |
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| 80 | %gm% add here afigure of the slope in i-direction |
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| 81 | |
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[9393] | 82 | \subsection{Slopes for tracer geopotential mixing in the $s$-coordinate} |
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[707] | 83 | |
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[11435] | 84 | In $s$-coordinates, geopotential mixing (\ie\ horizontal mixing) $r_1$ and $r_2$ are the slopes between |
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[10354] | 85 | the geopotential and computational surfaces. |
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[11543] | 86 | Their discrete formulation is found by locally solving \autoref{eq:TRA_ldf_iso} when |
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[10354] | 87 | the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform, |
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[11558] | 88 | \ie\ a linear function of $z_T$, the depth of a $T$-point. |
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[999] | 89 | %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} |
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[707] | 90 | |
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[10414] | 91 | \begin{equation} |
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[11543] | 92 | \label{eq:LDF_slp_geo} |
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[10414] | 93 | \begin{aligned} |
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| 94 | r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} |
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| 95 | \;\delta_{i+1/2}[z_t] |
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| 96 | &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] \ \ \ \\ |
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| 97 | r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} |
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| 98 | \;\delta_{j+1/2} [z_t] |
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| 99 | &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] \ \ \ \\ |
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| 100 | r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2} |
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| 101 | &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}] \\ |
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| 102 | r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2} |
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| 103 | &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}] |
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| 104 | \end{aligned} |
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[707] | 105 | \end{equation} |
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| 106 | |
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[11558] | 107 | %gm% caution I'm not sure the simplification was a good idea! |
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[707] | 108 | |
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[11537] | 109 | These slopes are computed once in \rou{ldf\_slp\_init} when \np{ln\_sco}\forcode{=.true.}, |
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[11558] | 110 | and either \np{ln\_traldf\_hor}\forcode{=.true.} or \np{ln\_dynldf\_hor}\forcode{=.true.}. |
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[999] | 111 | |
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[9393] | 112 | \subsection{Slopes for tracer iso-neutral mixing} |
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[9407] | 113 | \label{subsec:LDF_slp_iso} |
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[10414] | 114 | |
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[10354] | 115 | In iso-neutral mixing $r_1$ and $r_2$ are the slopes between the iso-neutral and computational surfaces. |
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| 116 | Their formulation does not depend on the vertical coordinate used. |
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| 117 | Their discrete formulation is found using the fact that the diffusive fluxes of |
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[11435] | 118 | locally referenced potential density (\ie\ $in situ$ density) vanish. |
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[11543] | 119 | So, substituting $T$ by $\rho$ in \autoref{eq:TRA_ldf_iso} and setting the diffusive fluxes in |
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[10354] | 120 | the three directions to zero leads to the following definition for the neutral slopes: |
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[707] | 121 | |
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[10414] | 122 | \begin{equation} |
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[11543] | 123 | \label{eq:LDF_slp_iso} |
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[10414] | 124 | \begin{split} |
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| 125 | r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} |
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| 126 | {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}} \\ |
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| 127 | r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]} |
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| 128 | {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}} \\ |
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| 129 | r_{1w} &= \frac{e_{3w}}{e_{1w}}\; |
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| 130 | \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}} |
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| 131 | {\delta_{k+1/2}[\rho]} \\ |
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| 132 | r_{2w} &= \frac{e_{3w}}{e_{2w}}\; |
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| 133 | \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}} |
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| 134 | {\delta_{k+1/2}[\rho]} |
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| 135 | \end{split} |
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[707] | 136 | \end{equation} |
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| 137 | |
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[1224] | 138 | %gm% rewrite this as the explanation is not very clear !!! |
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[11558] | 139 | %In practice, \autoref{eq:LDF_slp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \autoref{eq:LDF_slp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth. |
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[707] | 140 | |
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[11543] | 141 | %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{mcdougall_JPO87}, therefore in \autoref{eq:LDF_slp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters). |
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[707] | 142 | |
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[11558] | 143 | %In the $z$-coordinate, the derivative of the \autoref{eq:LDF_slp_iso} numerator is evaluated at the same depth \nocite{as what?} ($T$-level, which is the same as the $u$- and $v$-levels), so the $in situ$ density can be used for its evaluation. |
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[707] | 144 | |
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[11543] | 145 | As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:LDF_slp_iso} has to |
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[10354] | 146 | be evaluated at the same local pressure |
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| 147 | (which, in decibars, is approximated by the depth in meters in the model). |
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[11543] | 148 | Therefore \autoref{eq:LDF_slp_iso} cannot be used as such, |
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[10354] | 149 | but further transformation is needed depending on the vertical coordinate used: |
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[707] | 150 | |
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| 151 | \begin{description} |
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| 152 | |
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[10354] | 153 | \item[$z$-coordinate with full step: ] |
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[11543] | 154 | in \autoref{eq:LDF_slp_iso} the densities appearing in the $i$ and $j$ derivatives are taken at the same depth, |
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[10354] | 155 | thus the $in situ$ density can be used. |
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| 156 | This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, |
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[11123] | 157 | where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{mcdougall_JPO87} |
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[11558] | 158 | (see \autoref{subsec:TRA_bn2}). |
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[707] | 159 | |
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[10354] | 160 | \item[$z$-coordinate with partial step: ] |
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| 161 | this case is identical to the full step case except that at partial step level, |
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| 162 | the \emph{horizontal} density gradient is evaluated as described in \autoref{sec:TRA_zpshde}. |
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[707] | 163 | |
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[10354] | 164 | \item[$s$- or hybrid $s$-$z$- coordinate: ] |
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| 165 | in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if |
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[11537] | 166 | the Griffies scheme is used (\np{ln\_traldf\_triad}\forcode{=.true.}; |
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[11543] | 167 | see \autoref{apdx:TRIADS}). |
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[10354] | 168 | In other words, iso-neutral mixing will only be accurately represented with a linear equation of state |
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[11537] | 169 | (\np{ln\_seos}\forcode{=.true.}). |
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[11543] | 170 | In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:LDF_slp_iso} |
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[10354] | 171 | will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. |
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[707] | 172 | |
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[11558] | 173 | %gm% |
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[10354] | 174 | Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for |
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| 175 | the constraint on iso-neutral fluxes. |
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[11123] | 176 | Following \citet{griffies_bk04}, instead of specifying directly that there is a zero neutral diffusive flux of |
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[10354] | 177 | locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between |
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| 178 | the neutral direction diffusive fluxes of potential temperature and salinity: |
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[10414] | 179 | \[ |
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| 180 | \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S) |
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| 181 | \] |
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| 182 | % gm{ where vector F is ....} |
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[707] | 183 | |
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| 184 | This constraint leads to the following definition for the slopes: |
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| 185 | |
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[10414] | 186 | \[ |
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[11543] | 187 | % \label{eq:LDF_slp_iso2} |
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[10414] | 188 | \begin{split} |
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| 189 | r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac |
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| 190 | {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]} |
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| 191 | {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k} |
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| 192 | -\beta_u \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} } \\ |
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| 193 | r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac |
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| 194 | {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]} |
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| 195 | {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k} |
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| 196 | -\beta_v \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} } \\ |
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| 197 | r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac |
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| 198 | {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2} |
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| 199 | -\beta_w \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} } |
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| 200 | {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\ |
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| 201 | r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac |
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| 202 | {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2} |
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| 203 | -\beta_w \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} } |
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| 204 | {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\ |
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| 205 | \end{split} |
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| 206 | \] |
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[10354] | 207 | where $\alpha$ and $\beta$, the thermal expansion and saline contraction coefficients introduced in |
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| 208 | \autoref{subsec:TRA_bn2}, have to be evaluated at the three velocity points. |
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| 209 | In order to save computation time, they should be approximated by the mean of their values at $T$-points |
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| 210 | (for example in the case of $\alpha$: |
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| 211 | $\alpha_u=\overline{\alpha_T}^{i+1/2}$, $\alpha_v=\overline{\alpha_T}^{j+1/2}$ and |
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| 212 | $\alpha_w=\overline{\alpha_T}^{k+1/2}$). |
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[707] | 213 | |
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[10354] | 214 | Note that such a formulation could be also used in the $z$-coordinate and $z$-coordinate with partial steps cases. |
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[707] | 215 | |
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| 216 | \end{description} |
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| 217 | |
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[10354] | 218 | This implementation is a rather old one. |
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[11336] | 219 | It is similar to the one proposed by \citet{cox_OM87}, except for the background horizontal diffusion. |
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| 220 | Indeed, the \citet{cox_OM87} implementation of isopycnal diffusion in GFDL-type models requires |
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[10354] | 221 | a minimum background horizontal diffusion for numerical stability reasons. |
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| 222 | To overcome this problem, several techniques have been proposed in which the numerical schemes of |
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[11123] | 223 | the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}. |
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[11543] | 224 | Griffies's scheme is now available in \NEMO\ if \np{ln\_traldf\_triad}\forcode{ = .true.}; see \autoref{apdx:TRIADS}. |
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[11123] | 225 | Here, another strategy is presented \citep{lazar_phd97}: |
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[10354] | 226 | a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of |
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| 227 | grid point noise generated by the iso-neutral diffusion operator (\autoref{fig:LDF_ZDF1}). |
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| 228 | This allows an iso-neutral diffusion scheme without additional background horizontal mixing. |
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| 229 | This technique can be viewed as a diffusion operator that acts along large-scale |
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| 230 | (2~$\Delta$x) \gmcomment{2deltax doesnt seem very large scale} iso-neutral surfaces. |
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| 231 | The diapycnal diffusion required for numerical stability is thus minimized and its net effect on the flow is quite small when compared to the effect of an horizontal background mixing. |
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[707] | 232 | |
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[10354] | 233 | Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, |
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[11558] | 234 | contrary to the \citet{griffies.gnanadesikan.ea_JPO98} operator which has that property. |
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[707] | 235 | |
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| 236 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[10414] | 237 | \begin{figure}[!ht] |
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[11558] | 238 | \centering |
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[11561] | 239 | \includegraphics[width=0.66\textwidth]{Fig_LDF_ZDF1} |
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[11558] | 240 | \caption{Averaging procedure for isopycnal slope computation} |
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| 241 | \label{fig:LDF_ZDF1} |
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[10414] | 242 | \end{figure} |
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[707] | 243 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 244 | |
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[11558] | 245 | %There are three additional questions about the slope calculation. |
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| 246 | %First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes. |
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| 247 | %Second, numerical stability issues also require a bound on slopes. |
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[1224] | 248 | %Third, the question of boundary condition specified on slopes... |
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[707] | 249 | |
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| 250 | %from griffies: chapter 13.1.... |
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| 251 | |
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| 252 | |
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| 253 | |
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[11558] | 254 | % In addition and also for numerical stability reasons \citep{cox_OM87, griffies_bk04}, |
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| 255 | % the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly |
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| 256 | % to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the |
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[3294] | 257 | % surface motivates this flattening of isopycnals near the surface). |
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[707] | 258 | |
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[11123] | 259 | For numerical stability reasons \citep{cox_OM87, griffies_bk04}, the slopes must also be bounded by |
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[11336] | 260 | the namelist scalar \np{rn\_slpmax} (usually $1/100$) everywhere. |
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[10354] | 261 | This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to |
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| 262 | $1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean |
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| 263 | (the fact that the eddies "feel" the surface motivates this flattening of isopycnals near the surface). |
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[11336] | 264 | \colorbox{yellow}{The way slopes are tapered has be checked. Not sure that this is still what is actually done.} |
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[999] | 265 | |
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[707] | 266 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[10354] | 267 | \begin{figure}[!ht] |
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[11558] | 268 | \centering |
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[11561] | 269 | \includegraphics[width=0.66\textwidth]{Fig_eiv_slp} |
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[11558] | 270 | \caption[Vertical profile of the slope used for lateral mixing in the mixed layer]{ |
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| 271 | Vertical profile of the slope used for lateral mixing in the mixed layer: |
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| 272 | \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, |
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| 273 | which has to be adjusted at the surface boundary |
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| 274 | \ie\ it must tend to zero at the surface since there is no mixing across the air-sea interface: |
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| 275 | wall boundary condition). |
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| 276 | Nevertheless, |
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| 277 | the profile between the surface zero value and the interior iso-neutral one is unknown, |
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| 278 | and especially the value at the base of the mixed layer; |
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| 279 | \textit{(b)} profile of slope using a linear tapering of the slope near the surface and |
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| 280 | imposing a maximum slope of 1/100; |
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| 281 | \textit{(c)} profile of slope actually used in \NEMO: |
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| 282 | a linear decrease of the slope from zero at the surface to |
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| 283 | its ocean interior value computed just below the mixed layer. |
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| 284 | Note the huge change in the slope at the base of the mixed layer between |
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| 285 | \textit{(b)} and \textit{(c)}.} |
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| 286 | \label{fig:LDF_eiv_slp} |
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[10354] | 287 | \end{figure} |
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[707] | 288 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 289 | |
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[10354] | 290 | \colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.} |
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[707] | 291 | |
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[9393] | 292 | \subsection{Slopes for momentum iso-neutral mixing} |
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[707] | 293 | |
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[10354] | 294 | The iso-neutral diffusion operator on momentum is the same as the one used on tracers but |
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| 295 | applied to each component of the velocity separately |
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[11543] | 296 | (see \autoref{eq:DYN_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). |
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[10354] | 297 | The slopes between the surface along which the diffusion operator acts and the surface of computation |
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| 298 | ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and |
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| 299 | \textit{vw}- points for the $v$-component. |
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| 300 | They are computed from the slopes used for tracer diffusion, |
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[11543] | 301 | \ie\ \autoref{eq:LDF_slp_geo} and \autoref{eq:LDF_slp_iso}: |
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[707] | 302 | |
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[10414] | 303 | \[ |
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[11543] | 304 | % \label{eq:LDF_slp_dyn} |
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[10414] | 305 | \begin{aligned} |
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| 306 | &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ |
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| 307 | &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& r_{2t}\ &= \overline{r_{2v}}^{\,j} \\ |
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| 308 | &r_{1uw} = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ & r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\ |
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| 309 | &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&& r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\ |
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| 310 | \end{aligned} |
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| 311 | \] |
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[707] | 312 | |
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[10354] | 313 | The major issue remaining is in the specification of the boundary conditions. |
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| 314 | The same boundary conditions are chosen as those used for lateral diffusion along model level surfaces, |
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[11435] | 315 | \ie\ using the shear computed along the model levels and with no additional friction at the ocean bottom |
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[10354] | 316 | (see \autoref{sec:LBC_coast}). |
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[707] | 317 | |
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[11558] | 318 | |
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[6140] | 319 | % ================================================================ |
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| 320 | % Lateral Mixing Coefficients |
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| 321 | % ================================================================ |
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[11565] | 322 | \section[Lateral mixing coefficient (\forcode{nn_aht_ijk_t} \& \forcode{nn_ahm_ijk_t})]{Lateral mixing coefficient (\protect\np{nn\_aht\_ijk\_t} \& \protect\np{nn\_ahm\_ijk\_t})} |
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[9407] | 323 | \label{sec:LDF_coef} |
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[6140] | 324 | |
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[11558] | 325 | The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}. |
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[11336] | 326 | The way the mixing coefficients are set in the reference version can be described as follows: |
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[6140] | 327 | |
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[11565] | 328 | \subsection[Mixing coefficients read from file (\forcode{=-20, -30})]{ Mixing coefficients read from file (\protect\np{nn\_aht\_ijk\_t}\forcode{=-20, -30} \& \protect\np{nn\_ahm\_ijk\_t}\forcode{=-20, -30})} |
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[6140] | 329 | |
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[11558] | 330 | Mixing coefficients can be read from file if a particular geographical variation is needed. For example, in the ORCA2 global ocean model, |
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[11336] | 331 | the laplacian viscosity operator uses $A^l$~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and |
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[11558] | 332 | decreases linearly to $A^l$~= 2.10$^3$ m$^2$/s at the equator \citep{madec.delecluse.ea_JPO96, delecluse.madec_icol99}. |
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| 333 | Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of ORCA2 and ORCA05. |
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[11537] | 334 | The provided fields can either be 2d (\np{nn\_aht\_ijk\_t}\forcode{=-20}, \np{nn\_ahm\_ijk\_t}\forcode{=-20}) or 3d (\np{nn\_aht\_ijk\_t}\forcode{=-30}, \np{nn\_ahm\_ijk\_t}\forcode{=-30}). They must be given at U, V points for tracers and T, F points for momentum (see \autoref{tab:LDF_files}). |
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[6140] | 335 | |
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[11336] | 336 | %-------------------------------------------------TABLE--------------------------------------------------- |
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| 337 | \begin{table}[htb] |
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[11558] | 338 | \centering |
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| 339 | \begin{tabular}{|l|l|l|l|} |
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| 340 | \hline |
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| 341 | Namelist parameter & Input filename & dimensions & variable names \\ \hline |
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| 342 | \np{nn\_ahm\_ijk\_t}\forcode{=-20} & \forcode{eddy_viscosity_2D.nc } & $(i,j)$ & \forcode{ahmt_2d, ahmf_2d} \\ \hline |
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| 343 | \np{nn\_aht\_ijk\_t}\forcode{=-20} & \forcode{eddy_diffusivity_2D.nc } & $(i,j)$ & \forcode{ahtu_2d, ahtv_2d} \\ \hline |
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| 344 | \np{nn\_ahm\_ijk\_t}\forcode{=-30} & \forcode{eddy_viscosity_3D.nc } & $(i,j,k)$ & \forcode{ahmt_3d, ahmf_3d} \\ \hline |
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| 345 | \np{nn\_aht\_ijk\_t}\forcode{=-30} & \forcode{eddy_diffusivity_3D.nc } & $(i,j,k)$ & \forcode{ahtu_3d, ahtv_3d} \\ \hline |
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| 346 | \end{tabular} |
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| 347 | \caption{Description of expected input files if mixing coefficients are read from NetCDF files} |
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| 348 | \label{tab:LDF_files} |
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[11336] | 349 | \end{table} |
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| 350 | %-------------------------------------------------------------------------------------------------------------- |
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| 351 | |
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[11565] | 352 | \subsection[Constant mixing coefficients (\forcode{=0})]{ Constant mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=0} \& \protect\np{nn\_ahm\_ijk\_t}\forcode{=0})} |
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[11336] | 353 | |
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| 354 | If constant, mixing coefficients are set thanks to a velocity and a length scales ($U_{scl}$, $L_{scl}$) such that: |
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| 355 | |
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| 356 | \begin{equation} |
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[11543] | 357 | \label{eq:LDF_constantah} |
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[11336] | 358 | A_o^l = \left\{ |
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| 359 | \begin{aligned} |
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| 360 | & \frac{1}{2} U_{scl} L_{scl} & \text{for laplacian operator } \\ |
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| 361 | & \frac{1}{12} U_{scl} L_{scl}^3 & \text{for bilaplacian operator } |
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| 362 | \end{aligned} |
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| 363 | \right. |
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| 364 | \end{equation} |
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| 365 | |
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| 366 | $U_{scl}$ and $L_{scl}$ are given by the namelist parameters \np{rn\_Ud}, \np{rn\_Uv}, \np{rn\_Ld} and \np{rn\_Lv}. |
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| 367 | |
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[11565] | 368 | \subsection[Vertically varying mixing coefficients (\forcode{=10})]{Vertically varying mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=10} \& \protect\np{nn\_ahm\_ijk\_t}\forcode{=10})} |
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[11336] | 369 | |
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| 370 | In the vertically varying case, a hyperbolic variation of the lateral mixing coefficient is introduced in which |
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[11543] | 371 | the surface value is given by \autoref{eq:LDF_constantah}, the bottom value is 1/4 of the surface value, |
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[11336] | 372 | and the transition takes place around z=500~m with a width of 200~m. |
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| 373 | This profile is hard coded in module \mdl{ldfc1d\_c2d}, but can be easily modified by users. |
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| 374 | |
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[11565] | 375 | \subsection[Mesh size dependent mixing coefficients (\forcode{=20})]{Mesh size dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=20} \& \protect\np{nn\_ahm\_ijk\_t}\forcode{=20})} |
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[11336] | 376 | |
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| 377 | In that case, the horizontal variation of the eddy coefficient depends on the local mesh size and |
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[10354] | 378 | the type of operator used: |
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[10414] | 379 | \begin{equation} |
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[11543] | 380 | \label{eq:LDF_title} |
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[10414] | 381 | A_l = \left\{ |
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| 382 | \begin{aligned} |
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[11386] | 383 | & \frac{1}{2} U_{scl} \max(e_1,e_2) & \text{for laplacian operator } \\ |
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| 384 | & \frac{1}{12} U_{scl} \max(e_1,e_2)^{3} & \text{for bilaplacian operator } |
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[10414] | 385 | \end{aligned} |
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| 386 | \right. |
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[6140] | 387 | \end{equation} |
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[11386] | 388 | where $U_{scl}$ is a user defined velocity scale (\np{rn\_Ud}, \np{rn\_Uv}). |
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[10354] | 389 | This variation is intended to reflect the lesser need for subgrid scale eddy mixing where |
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| 390 | the grid size is smaller in the domain. |
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[11123] | 391 | It was introduced in the context of the DYNAMO modelling project \citep{willebrand.barnier.ea_PO01}. |
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[11336] | 392 | Note that such a grid scale dependance of mixing coefficients significantly increases the range of stability of |
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| 393 | model configurations presenting large changes in grid spacing such as global ocean models. |
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[10354] | 394 | Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to |
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[11543] | 395 | large coefficient compare to the smallest grid size (see \autoref{sec:TD_forward_imp}), |
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[10354] | 396 | especially when using a bilaplacian operator. |
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[6140] | 397 | |
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[11336] | 398 | \colorbox{yellow}{CASE \np{nn\_aht\_ijk\_t} = 21 to be added} |
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[6140] | 399 | |
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[11565] | 400 | \subsection[Mesh size and depth dependent mixing coefficients (\forcode{=30})]{Mesh size and depth dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=30} \& \protect\np{nn\_ahm\_ijk\_t}\forcode{=30})} |
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[6140] | 401 | |
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[11336] | 402 | The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases above, |
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[11435] | 403 | \ie\ a hyperbolic tangent variation with depth associated with a grid size dependence of |
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[11558] | 404 | the magnitude of the coefficient. |
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[6140] | 405 | |
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[11565] | 406 | \subsection[Velocity dependent mixing coefficients (\forcode{=31})]{Flow dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=31} \& \protect\np{nn\_ahm\_ijk\_t}\forcode{=31})} |
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[11336] | 407 | In that case, the eddy coefficient is proportional to the local velocity magnitude so that the Reynolds number $Re = \lvert U \rvert e / A_l$ is constant (and here hardcoded to $12$): |
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| 408 | \colorbox{yellow}{JC comment: The Reynolds is effectively set to 12 in the code for both operators but shouldn't it be 2 for Laplacian ?} |
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[6140] | 409 | |
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[11336] | 410 | \begin{equation} |
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[11543] | 411 | \label{eq:LDF_flowah} |
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[11336] | 412 | A_l = \left\{ |
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| 413 | \begin{aligned} |
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| 414 | & \frac{1}{12} \lvert U \rvert e & \text{for laplacian operator } \\ |
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[11558] | 415 | & \frac{1}{12} \lvert U \rvert e^3 & \text{for bilaplacian operator } |
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[11336] | 416 | \end{aligned} |
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| 417 | \right. |
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| 418 | \end{equation} |
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[6140] | 419 | |
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[11565] | 420 | \subsection[Deformation rate dependent viscosities (\forcode{nn_ahm_ijk_t=32})]{Deformation rate dependent viscosities (\protect\np{nn\_ahm\_ijk\_t}\forcode{=32})} |
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[11336] | 421 | |
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[11558] | 422 | This option refers to the \citep{smagorinsky_MW63} scheme which is here implemented for momentum only. Smagorinsky chose as a |
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[11336] | 423 | characteristic time scale $T_{smag}$ the deformation rate and for the lengthscale $L_{smag}$ the maximum wavenumber possible on the horizontal grid, e.g.: |
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| 424 | |
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| 425 | \begin{equation} |
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[11543] | 426 | \label{eq:LDF_smag1} |
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[11336] | 427 | \begin{split} |
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| 428 | T_{smag}^{-1} & = \sqrt{\left( \partial_x u - \partial_y v\right)^2 + \left( \partial_y u + \partial_x v\right)^2 } \\ |
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| 429 | L_{smag} & = \frac{1}{\pi}\frac{e_1 e_2}{e_1 + e_2} |
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| 430 | \end{split} |
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| 431 | \end{equation} |
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| 432 | |
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| 433 | Introducing a user defined constant $C$ (given in the namelist as \np{rn\_csmc}), one can deduce the mixing coefficients as follows: |
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| 434 | |
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| 435 | \begin{equation} |
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[11543] | 436 | \label{eq:LDF_smag2} |
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[11336] | 437 | A_{smag} = \left\{ |
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| 438 | \begin{aligned} |
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| 439 | & C^2 T_{smag}^{-1} L_{smag}^2 & \text{for laplacian operator } \\ |
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[11558] | 440 | & \frac{C^2}{8} T_{smag}^{-1} L_{smag}^4 & \text{for bilaplacian operator } |
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[11336] | 441 | \end{aligned} |
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| 442 | \right. |
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| 443 | \end{equation} |
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| 444 | |
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[11543] | 445 | For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:TD_forward_imp}) so that: |
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[11336] | 446 | \begin{equation} |
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[11543] | 447 | \label{eq:LDF_smag3} |
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[11336] | 448 | \begin{aligned} |
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| 449 | & C_{min} \frac{1}{2} \lvert U \rvert e < A_{smag} < C_{max} \frac{e^2}{ 8\rdt} & \text{for laplacian operator } \\ |
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[11558] | 450 | & C_{min} \frac{1}{12} \lvert U \rvert e^3 < A_{smag} < C_{max} \frac{e^4}{64 \rdt} & \text{for bilaplacian operator } |
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[11336] | 451 | \end{aligned} |
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| 452 | \end{equation} |
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| 453 | |
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| 454 | where $C_{min}$ and $C_{max}$ are adimensional namelist parameters given by \np{rn\_minfac} and \np{rn\_maxfac} respectively. |
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| 455 | |
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| 456 | \subsection{About space and time varying mixing coefficients} |
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| 457 | |
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[6140] | 458 | The following points are relevant when the eddy coefficient varies spatially: |
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| 459 | |
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[10354] | 460 | (1) the momentum diffusion operator acting along model level surfaces is written in terms of curl and |
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[11543] | 461 | divergent components of the horizontal current (see \autoref{subsec:MB_ldf}). |
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[10354] | 462 | Although the eddy coefficient could be set to different values in these two terms, |
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[11558] | 463 | this option is not currently available. |
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[6140] | 464 | |
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[10354] | 465 | (2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of |
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| 466 | the horizontal divergence for operators acting along model-surfaces are no longer satisfied |
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[11543] | 467 | (\autoref{sec:INVARIANTS_dynldf_properties}). |
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[6140] | 468 | |
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[11336] | 469 | % ================================================================ |
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| 470 | % Eddy Induced Mixing |
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| 471 | % ================================================================ |
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[11565] | 472 | \section[Eddy induced velocity (\forcode{ln_ldfeiv})]{Eddy induced velocity (\protect\np{ln\_ldfeiv})} |
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[6140] | 473 | |
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[11336] | 474 | \label{sec:LDF_eiv} |
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[6140] | 475 | |
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[11336] | 476 | %--------------------------------------------namtra_eiv--------------------------------------------------- |
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[6140] | 477 | |
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[11558] | 478 | \begin{listing} |
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| 479 | \nlst{namtra_eiv} |
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[11567] | 480 | \caption{\forcode{&namtra_eiv}} |
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[11558] | 481 | \label{lst:namtra_eiv} |
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| 482 | \end{listing} |
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[6140] | 483 | |
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[11336] | 484 | %-------------------------------------------------------------------------------------------------------------- |
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[6140] | 485 | |
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[707] | 486 | |
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[6289] | 487 | %%gm from Triad appendix : to be incorporated.... |
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| 488 | \gmcomment{ |
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[10354] | 489 | Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}. |
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| 490 | If none of the keys \key{traldf\_cNd}, N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and |
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| 491 | GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and \np{rn\_aeiv\_0}. |
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| 492 | If 2D-varying coefficients are set with \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal |
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[10414] | 493 | scale factor according to \autoref{eq:title} |
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| 494 | \footnote{ |
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[10354] | 495 | Except in global ORCA $0.5^{\circ}$ runs with \key{traldf\_eiv}, |
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| 496 | where $A_l$ is set like $A_e$ but with a minimum vale of $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$ |
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| 497 | }. |
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| 498 | In idealised setups with \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} is set in |
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| 499 | the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is instead set from |
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[10414] | 500 | the Held-Larichev parameterisation |
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| 501 | \footnote{ |
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[10354] | 502 | In this case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further reduced by a factor $|f/f_{20}|$, |
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| 503 | where $f_{20}$ is the value of $f$ at $20^{\circ}$~N |
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| 504 | } (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored unless it is zero. |
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[6289] | 505 | } |
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| 506 | |
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[11537] | 507 | When \citet{gent.mcwilliams_JPO90} diffusion is used (\np{ln\_ldfeiv}\forcode{=.true.}), |
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[10354] | 508 | an eddy induced tracer advection term is added, |
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| 509 | the formulation of which depends on the slopes of iso-neutral surfaces. |
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| 510 | Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, |
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[11543] | 511 | \ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinates, |
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| 512 | and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $s$-coordinates. |
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[11336] | 513 | |
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[11558] | 514 | If isopycnal mixing is used in the standard way, \ie\ \np{ln\_traldf\_triad}\forcode{=.false.}, the eddy induced velocity is given by: |
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[10414] | 515 | \begin{equation} |
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[11543] | 516 | \label{eq:LDF_eiv} |
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[10414] | 517 | \begin{split} |
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| 518 | u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ |
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| 519 | v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\ |
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| 520 | w^* & = \frac{1}{e_{1w}e_{2w}}\; \left\{ \delta_i \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right] + \delta_j \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right] \right\} \\ |
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| 521 | \end{split} |
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[707] | 522 | \end{equation} |
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[11558] | 523 | where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{nn\_aei\_ijk\_t} \nam{tra\_eiv} namelist parameter. |
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[11336] | 524 | The three components of the eddy induced velocity are computed in \rou{ldf\_eiv\_trp} and |
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| 525 | added to the eulerian velocity in \rou{tra\_adv} where tracer advection is performed. |
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[10354] | 526 | This has been preferred to a separate computation of the advective trends associated with the eiv velocity, |
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| 527 | since it allows us to take advantage of all the advection schemes offered for the tracers |
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| 528 | (see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection scheme as in |
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[11123] | 529 | previous releases of OPA \citep{madec.delecluse.ea_NPM98}. |
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[10354] | 530 | This is particularly useful for passive tracers where \emph{positivity} of the advection scheme is of |
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[11558] | 531 | paramount importance. |
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[707] | 532 | |
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[10354] | 533 | At the surface, lateral and bottom boundaries, the eddy induced velocity, |
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[11558] | 534 | and thus the advective eddy fluxes of heat and salt, are set to zero. |
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| 535 | The value of the eddy induced mixing coefficient and its space variation is controlled in a similar way as for lateral mixing coefficient described in the preceding subsection (\np{nn\_aei\_ijk\_t}, \np{rn\_Ue}, \np{rn\_Le} namelist parameters). |
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[11336] | 536 | \colorbox{yellow}{CASE \np{nn\_aei\_ijk\_t} = 21 to be added} |
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[707] | 537 | |
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[11543] | 538 | In case of setting \np{ln\_traldf\_triad}\forcode{ = .true.}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:TRIADS}. |
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[11336] | 539 | |
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| 540 | % ================================================================ |
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| 541 | % Mixed layer eddies |
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| 542 | % ================================================================ |
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[11565] | 543 | \section[Mixed layer eddies (\forcode{ln_mle})]{Mixed layer eddies (\protect\np{ln\_mle})} |
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[11336] | 544 | \label{sec:LDF_mle} |
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| 545 | |
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| 546 | %--------------------------------------------namtra_eiv--------------------------------------------------- |
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| 547 | |
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[11558] | 548 | \begin{listing} |
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| 549 | \nlst{namtra_mle} |
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[11567] | 550 | \caption{\forcode{&namtra_mle}} |
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[11558] | 551 | \label{lst:namtra_mle} |
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| 552 | \end{listing} |
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[11336] | 553 | |
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| 554 | %-------------------------------------------------------------------------------------------------------------- |
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| 555 | |
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[11537] | 556 | If \np{ln\_mle}\forcode{=.true.} in \nam{tra\_mle} namelist, a parameterization of the mixing due to unresolved mixed layer instabilities is activated (\citet{foxkemper.ferrari_JPO08}). Additional transport is computed in \rou{ldf\_mle\_trp} and added to the eulerian transport in \rou{tra\_adv} as done for eddy induced advection. |
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[11336] | 557 | |
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| 558 | \colorbox{yellow}{TBC} |
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| 559 | |
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[10414] | 560 | \biblio |
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[707] | 561 | |
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[10442] | 562 | \pindex |
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| 563 | |
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[9363] | 564 | \end{document} |
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