Changeset 7598
- Timestamp:
- 2017-01-23T14:13:19+01:00 (7 years ago)
- Location:
- branches/2016/dev_merge_2016/NEMOGCM/CONFIG/WAD_TEST_CASES/MY_DOCS
- Files:
-
- 2 edited
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branches/2016/dev_merge_2016/NEMOGCM/CONFIG/WAD_TEST_CASES/MY_DOCS/WAD_doc.tex
r7547 r7598 115 115 116 116 Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and 117 $\mathrm{flxv}^{(m)}$. The iteration is initialised by setting 117 $\mathrm{flxv}^{(m)}$. Then the adjustment is achieved by seeking a set of coefficients, 118 $\mathrm{zcoef}_{i,j}^{(m)}$ such that: 119 120 \begin{equation} \label{dyn_wd_continuity_coef} 121 \begin{split} 122 \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\ 123 \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j} 124 \end{split} 125 \end{equation} 126 127 where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around 128 cells that would otherwise dry. 129 130 The iteration is initialised by setting 118 131 119 132 \begin{equation} \label{dyn_wd_zzflx_initial} … … 130 143 \end{equation} 131 144 132 Where this is the case each of the fluxes out of this $(i,j)$ cell are multiplied by the 133 factor $\mathrm{zcoef}_{i,j}$:134 135 \begin{equation} \label{dyn_wd_ continuity_coef}145 Rearranging (\ref{dyn_wd_continuity_if}) we can obtain an expression for the maximum 146 outward flux that can be allowed and still maintain the minimum wet depth: 147 148 \begin{equation} \label{dyn_wd_max_flux} 136 149 \begin{split} 137 \mathrm{z coef}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2}) \frac{e_1 e_2}{\Delta t} \phantom{]} \\138 \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(m)}_{i,j} }150 \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{]} \\ 151 \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] 139 152 \end{split} 140 \end{equation} 141 142 Note that the flux across the ``eastern'' face of the $(i,j)$th cell is only updated at 143 the $m+1$th iteration if that flux at the $m$th iteration is out of the $(i,j)$th cell. If 144 that is the case then the flux across that face is into the $(i+1,j)$ cell and that flux 145 will not be updated by the calculation for the $(i+1,j)$th cell. In this sense the updates 146 to the fluxes across the faces of the cells do not ``compete'' (they do not over-write 147 each other) and one would expect the scheme to converge relatively quickly. The scheme is 148 also flux based so conserves mass. 153 \end{equation} 154 155 Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\it [Q: Why is 156 this necessary/desirable?]}. Substituting from (\ref{dyn_wd_continuity_coef}) gives an 157 expression for the coefficient needed to multiply the outward flux at this cell in order 158 to avoid drying. 159 160 \begin{equation} \label{dyn_wd_continuity_nxtcoef} 161 \begin{split} 162 \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{]} \\ 163 \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} } 164 \end{split} 165 \end{equation} 166 167 Only the outward flux components are altered but, of course, outward fluxes from one cell 168 are inward fluxes to adjacent cells and the balance in these cells may need subsequent 169 adjustment; hence the iterative nature of this scheme. Note, for example, that the flux 170 across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration 171 if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then 172 the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by 173 the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across 174 the faces of the cells do not ``compete'' (they do not over-write each other) and one 175 would expect the scheme to converge relatively quickly. The scheme is also flux based so 176 conserves mass. 149 177 150 178 The ROMS scheme to prevent drying out of a cell is somewhat simpler. It specifies that if 151 179 a tracer cell is dry (the water depth is less than $\mathrm{rn\_wdmin1}$) on the backward 152 180 timestep, $t_e$, then any outward flux through its cell faces should be set to zero. This 153 scheme has a clear physical rationale. It has not yet been implemented within NEMO but it154 could be. One objection to the ROMS scheme is that it introduces a spurious step function 155 in the flux out of a cell as the water depth in the cell passes through the ``critical'' 156 value $\mathrm{rn\_wdmin1}$. One might replace this step function with a smoother function 157 of the water depth in the cell from which the flux originates.181 scheme has a clear physical rationale. This scheme is equivalent to setting 182 $\mathrm{zcoef}^{(m+1)}_{i,j}$ to $0.0$ whenever a cell is at risk of drying. One 183 objection to the ROMS scheme is that it introduces a spurious step function in the flux 184 out of a cell as the water depth in the cell passes through the ``critical'' value 185 $\mathrm{rn\_wdmin1}$. 158 186 159 187 %----------------------------------------------------------------------------------------
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