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Ignore:
Timestamp:
2017-01-23T14:13:19+01:00 (8 years ago)
Message:

Branch 2016/dev_merge_2016. Update Wetting and drying documentation (temporary location)

File:
1 edited

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Unmodified
 r7547 Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and $\mathrm{flxv}^{(m)}$. The iteration is initialised by setting $\mathrm{flxv}^{(m)}$.  Then the adjustment is achieved by seeking a set of coefficients, $\mathrm{zcoef}_{i,j}^{(m)}$ such that: \label{dyn_wd_continuity_coef} \begin{split} \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\ \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j} \end{split} where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around cells that would otherwise dry. The iteration is initialised by setting \label{dyn_wd_zzflx_initial} Where this is the case each of the fluxes out of this $(i,j)$ cell are multiplied by the factor $\mathrm{zcoef}_{i,j}$: \label{dyn_wd_continuity_coef} Rearranging (\ref{dyn_wd_continuity_if}) we can obtain an expression for the maximum outward flux that can be allowed and still maintain the minimum wet depth: \label{dyn_wd_max_flux} \begin{split} \mathrm{zcoef}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2})  \frac{e_1 e_2}{\Delta t} \phantom{]} \\ \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(m)}_{i,j} } \mathrm{zzflxp}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2})  \frac{e_1 e_2}{\Delta t} \phantom{]} \\ \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \end{split} Note that the flux across the eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across the faces of the cells do not compete'' (they do not over-write each other) and one would expect the scheme to converge relatively quickly. The scheme is also flux based so conserves mass. Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\it [Q: Why is this necessary/desirable?]}. Substituting from (\ref{dyn_wd_continuity_coef}) gives an expression for the coefficient needed to multiply the outward flux at this cell in order to avoid drying. \label{dyn_wd_continuity_nxtcoef} \begin{split} \mathrm{zcoef}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2})  \frac{e_1 e_2}{\Delta t} \phantom{]} \\ \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} } \end{split} Only the outward flux components are altered but, of course, outward fluxes from one cell are inward fluxes to adjacent cells and the balance in these cells may need subsequent adjustment; hence the iterative nature of this scheme.  Note, for example, that the flux across the eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across the faces of the cells do not compete'' (they do not over-write each other) and one would expect the scheme to converge relatively quickly. The scheme is also flux based so conserves mass. The ROMS scheme to prevent drying out of a cell is somewhat simpler. It specifies that if a tracer cell is dry (the water depth is less than $\mathrm{rn\_wdmin1}$) on the backward timestep, $t_e$, then any outward flux through its cell faces should be set to zero. This scheme has a clear physical rationale.  It has not yet been implemented within NEMO but it could be. One objection to the ROMS scheme is that it introduces a spurious step function in the flux out of a cell as the water depth in the cell passes through the critical'' value $\mathrm{rn\_wdmin1}$. One might replace this step function with a smoother function of the water depth in the cell from which the flux originates. scheme has a clear physical rationale. This scheme is equivalent to setting $\mathrm{zcoef}^{(m+1)}_{i,j}$ to $0.0$ whenever a cell is at risk of drying.  One objection to the ROMS scheme is that it introduces a spurious step function in the flux out of a cell as the water depth in the cell passes through the critical'' value $\mathrm{rn\_wdmin1}$. %----------------------------------------------------------------------------------------