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Changeset 10499 for NEMO/trunk/tests/WAD/MY_DOCS/WAD_doc.tex – NEMO

Ignore:
Timestamp:
2019-01-10T16:12:24+01:00 (5 years ago)
Author:
deazer
Message:

Fix ticket #2154

File:
1 edited

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  • NEMO/trunk/tests/WAD/MY_DOCS/WAD_doc.tex

    r9024 r10499  
    2525% Wetting and drying  
    2626% ================================================================ 
    27 \section{Wetting and drying } 
    28 \label{DYN_wetdry} 
    29  
    30 There are two main options for wetting and drying code (wd):  
    31 (a) an iterative limiter (il) and (b) a directional limiter (dl).  
    32 The directional limiter is based on the scheme developed by \cite{WarnerEtal13} for ROMS 
    33 which was in turn based on ideas developed for POM by \cite{Oey06}. The iterative 
    34 limiter is a new scheme.  The iterative limiter is activated by setting $\mathrm{ln\_wd\_il} = \mathrm{.true.}$ 
    35 and $\mathrm{ln\_wd\_dl} = \mathrm{.false.}$. The directional limiter is activated 
    36 by setting $\mathrm{ln\_wd\_dl} = \mathrm{.true.}$ and $\mathrm{ln\_wd\_il} = \mathrm{.false.}$. 
    37  
    38 \namdisplay{nam_wad} 
    39  
    40 The following terminology is used. The depth of the topography (positive downwards) 
    41 at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the NEMO code. 
    42 The height of the free surface (positive upwards) is denoted by $ \mathrm{ssh}$. Given the sign 
    43 conventions used, the water depth, $h$, is the height of the free surface plus the depth of the 
    44 topography (i.e. $\mathrm{ssh} + \mathrm{ht\_wd}$). 
    45  
    46 Both wd schemes take all points in the domain below a land elevation of $\mathrm{rn\_wdld}$ to be 
    47 covered by water. They require the topography specified with a model 
    48 configuration to have negative depths at points where the land is higher than the 
    49 topography's reference sea-level. The vertical grid in NEMO is normally computed relative to an 
    50 initial state with zero sea surface height elevation.  
    51 The user can choose to compute the vertical grid and heights in the model relative to  
    52 a non-zero reference height for the free surface. This choice affects the calculation of the metrics and depths 
    53 (i.e. the $\mathrm{e3t\_0, ht\_0}$ etc. arrays).  
    54  
    55 Points where the water depth is less than $\mathrm{rn\_wdmin1}$ are interpreted as ``dry''.  
    56 $\mathrm{rn\_wdmin1}$ is usually chosen to be of order $0.05$m but extreme topographies  
    57 with very steep slopes require larger values for normal choices of time-step.  
    58  
    59 Both versions of the code have been tested in six test cases provided in the WAD\_TEST\_CASES configuration 
    60 and in ``realistic'' configurations covering parts of the north-west European shelf.  
    61 All these configurations have used pure sigma coordinates. It is expected that 
    62 the wetting and drying code will work in domains with more general s-coordinates provided 
    63 the coordinates are pure sigma in the region where wetting and drying actually occurs.   
    64  
    65 The next sub-section descrbies the directional limiter and the following sub-section the iterative limiter.  
    66 The final sub-section covers some additional considerations that are relevant to both schemes.  
    67  
    68 %----------------------------------------------------------------------------------------- 
    69 %   Iterative limiters 
    70 %----------------------------------------------------------------------------------------- 
    71 \subsection   [Directional limiter (\textit{wet\_dry})] 
    72          {Directional limiter (\mdl{wet\_dry})} 
    73 \label{DYN_wd_directional_limiter} 
    74  
    75 The principal idea of the directional limiter is that  
    76 water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than rn\_wdmin1). 
    77  
    78 All the changes associated with this option are made to the barotropic solver for the non-linear  
    79 free surface code within dynspg\_ts.  
    80 On each barotropic sub-step the scheme determines the direction of the flow across each face of all the tracer cells 
    81 and sets the flux across the face to zero when the flux is from a dry tracer cell. This prevents cells 
    82 whose depth is rn\_wdmin1 or less from drying out further. The scheme does not force $h$ (the water depth) at tracer cells 
    83 to be at least the minimum depth and hence is able to conserve mass / volume.  
    84  
    85 The flux across each $u$-face of a tracer cell is multiplied by a factor zuwdmask (an array which depends on ji and jj).  
    86 If the user sets ln\_wd\_dl\_ramp = .False. then zuwdmask is 1 when the 
    87 flux is from a cell with water depth greater than rn\_wdmin1 and 0 otherwise. If the user sets 
    88 ln\_wd\_dl\_ramp = .True. the flux across the face is ramped down as the water depth decreases 
    89 from 2 * rn\_wdmin1 to rn\_wdmin1. The use of this ramp reduced grid-scale noise in idealised test cases.  
    90  
    91 At the point where the flux across a $u$-face is multiplied by zuwdmask , we have chosen 
    92 also to multiply the corresponding velocity on the ``now'' step at that face by zuwdmask. We could have 
    93 chosen not to do that and to allow fairly large velocities to occur in these ``dry'' cells.  
    94 The rationale for setting the velocity to zero is that it is the momentum equations that are being solved 
    95 and the total momentum of the upstream cell (treating it as a finite volume) should be considered 
    96 to be its depth times its velocity. This depth is considered to be zero at ``dry'' $u$-points consistent with its  
    97 treatment in the calculation of the flux of mass across the cell face.          
    98  
    99 \cite{WarnerEtal13} state that in their scheme the velocity masks at the cell faces for the baroclinic  
    100 timesteps are set to 0 or 1 depending on whether the average of the masks over the barotropic sub-steps is respectively less than  
    101 or greater than 0.5. That scheme does not conserve tracers in integrations started from constant tracer 
    102 fields (tracers independent of $x$, $y$ and $z$). Our scheme conserves constant tracers because 
    103 the velocities used at the tracer cell faces on the baroclinic timesteps are carefully calculated by dynspg\_ts 
    104 to equal their mean value during the barotropic steps. If the user sets ln\_wd\_dl\_bc = .True., the 
    105 baroclinic velocities are also multiplied by a suitably weighted average of zuwdmask.       
    106       
    107 %----------------------------------------------------------------------------------------- 
    108 %   Iterative limiters 
    109 %----------------------------------------------------------------------------------------- 
    110 \subsection   [Iterative limiter (\textit{wet\_dry})] 
    111          {Iterative limiter (\mdl{wet\_dry})} 
    112 \label{DYN_wd_iterative_limiter} 
    113  
    114 \subsubsection [Iterative flux limiter (\textit{wet\_dry})] 
    115          {Iterative flux limiter (\mdl{wet\_dry})} 
    116 \label{DYN_wd_il_spg_limiter} 
    117  
    118 The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry'' 
    119 or may become dry within the next time-step using an iterative method.  
    120  
    121 The flux limiter for the barotropic flow (devised by Hedong Liu) can be understood as follows:  
    122  
    123 The continuity equation for the total water depth in a column  
    124 \begin{equation} \label{dyn_wd_continuity} 
    125  \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 . 
    126 \end{equation}  
    127 can be written in discrete form  as   
    128  
    129 \begin{align} \label{dyn_wd_continuity_2} 
    130 \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) )  
    131 &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j}  + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\ 
    132 &= \mathrm{zzflx}_{i,j} . 
    133 \end{align}  
    134  
    135 In the above $h$ is the depth of the water in the column at point $(i,j)$, 
    136 $\mathrm{flxu}_{i+1,j}$ is the flux out of the ``eastern'' face of the cell and 
    137 $\mathrm{flxv}_{i,j+1}$ the flux out of the ``northern'' face of the cell; $t_{n+1}$ is 
    138 the new timestep, $t_e$ is the old timestep (either $t_b$ or $t_n$) and $ \Delta t = 
    139 t_{n+1} - t_e$; $e_1 e_2$ is the area of the tracer cells centred at $(i,j)$ and 
    140 $\mathrm{zzflx}$ is the sum of the fluxes through all the faces. 
    141  
    142 The flux limiter splits the flux $\mathrm{zzflx}$ into fluxes that are out of the cell 
    143 (zzflxp) and fluxes that are into the cell (zzflxn).  Clearly 
    144  
    145 \begin{equation} \label{dyn_wd_zzflx_p_n_1} 
    146 \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} .   
    147 \end{equation}  
    148  
    149 The flux limiter iteratively adjusts the fluxes $\mathrm{flxu}$ and $\mathrm{flxv}$ until 
    150 none of the cells will ``dry out''. To be precise the fluxes are limited until none of the 
    151 cells has water depth less than $\mathrm{rn\_wdmin1}$ on step $n+1$. 
    152  
    153 Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and 
    154 $\mathrm{flxv}^{(m)}$.  Then the adjustment is achieved by seeking a set of coefficients, 
    155 $\mathrm{zcoef}_{i,j}^{(m)}$ such that: 
    156  
    157 \begin{equation} \label{dyn_wd_continuity_coef} 
    158 \begin{split} 
    159 \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\ 
    160 \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j} 
    161 \end{split} 
    162 \end{equation}  
    163   
    164 where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around 
    165 cells that would otherwise dry. 
    166  
    167 The iteration is initialised by setting 
    168  
    169 \begin{equation} \label{dyn_wd_zzflx_initial} 
    170 \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad  \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} .  
    171 \end{equation}  
    172  
    173 The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the 
    174 cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell 
    175 times the timestep divided by the cell area. Using (\ref{dyn_wd_continuity_2}) this 
    176 condition is 
    177  
    178 \begin{equation} \label{dyn_wd_continuity_if} 
    179 h_{i,j}(t_e)  - \mathrm{rn\_wdmin1} <  \frac{\Delta t}{e_1 e_2} ( \mathrm{zzflxp}^{(m)}_{i,j} + \mathrm{zzflxn}^{(m)}_{i,j} ) . 
    180 \end{equation}  
    181  
    182 Rearranging (\ref{dyn_wd_continuity_if}) we can obtain an expression for the maximum 
    183 outward flux that can be allowed and still maintain the minimum wet depth: 
    184  
    185 \begin{equation} \label{dyn_wd_max_flux} 
    186 \begin{split} 
    187 \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{]} \\ 
    188 \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] 
    189 \end{split} 
    190 \end{equation} 
    191  
    192 Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\it [Q: Why is 
    193 this necessary/desirable?]}. Substituting from (\ref{dyn_wd_continuity_coef}) gives an 
    194 expression for the coefficient needed to multiply the outward flux at this cell in order 
    195 to avoid drying.  
    196  
    197 \begin{equation} \label{dyn_wd_continuity_nxtcoef} 
    198 \begin{split} 
    199 \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{]} \\ 
    200 \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} }  
    201 \end{split} 
    202 \end{equation}  
    203  
    204 Only the outward flux components are altered but, of course, outward fluxes from one cell 
    205 are inward fluxes to adjacent cells and the balance in these cells may need subsequent 
    206 adjustment; hence the iterative nature of this scheme.  Note, for example, that the flux 
    207 across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration 
    208 if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then 
    209 the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by 
    210 the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across 
    211 the faces of the cells do not ``compete'' (they do not over-write each other) and one 
    212 would expect the scheme to converge relatively quickly. The scheme is flux based so 
    213 conserves mass. It also conserves constant tracers for the same reason that the  
    214 directional limiter does.   
    215  
    216  
    217 %---------------------------------------------------------------------------------------- 
    218 %      Surface pressure gradients 
    219 %---------------------------------------------------------------------------------------- 
    220 \subsubsection   [Modification of surface pressure gradients (\textit{dynhpg})] 
    221          {Modification of surface pressure gradients (\mdl{dynhpg})} 
    222 \label{DYN_wd_il_spg} 
    223  
    224 At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the 
    225 topography is sloping at these points the sea-surface will have a similar slope and there 
    226 will hence be very large horizontal pressure gradients at these points. The WAD modifies 
    227 the magnitude but not the sign of the surface pressure gradients (zhpi and zhpj) at such 
    228 points by mulitplying them by positive factors (zcpx and zcpy respectively) that lie 
    229 between $0$ and $1$. 
    230  
    231 We describe how the scheme works for the ``eastward'' pressure gradient, zhpi, calculated 
    232 at the $(i,j)$th $u$-point. The scheme uses the ht\_wd depths and surface heights at the 
    233 neighbouring $(i+1,j)$ and $(i,j)$ tracer points.  zcpx is calculated using two logicals 
    234 variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid 
    235 column.  The three possible combinations are illustrated in figure \ref{Fig_WAD_dynhpg}. 
    236 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    237 \begin{figure}[!ht] \begin{center} 
    238 \includegraphics[width=0.8\textwidth]{Fig_WAD_dynhpg} 
    239 \caption{ \label{Fig_WAD_dynhpg} 
    240 Illustrations of the three possible combinations of the logical variables controlling the  
    241 limiting of the horizontal pressure gradient in wetting and drying regimes} 
    242 \end{center}\end{figure} 
    243 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    244  
    245 The first logical, $\mathrm{ll\_tmp1}$, is set to true if and only if the water depth at 
    246 both neighbouring points is greater than $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ and 
    247 the minimum height of the sea surface at the two points is greater than the maximum height 
    248 of the topography at the two points: 
    249  
    250 \begin{equation} \label{dyn_ll_tmp1} 
    251 \begin{split} 
    252 \mathrm{ll\_tmp1}  = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\ 
    253                      & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\  .and.} \\ 
    254 & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\ 
    255 & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\ 
    256 & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 } 
    257 \end{split} 
    258 \end{equation}  
    259  
    260 The second logical, $\mathrm{ll\_tmp2}$, is set to true if and only if the maximum height 
    261 of the sea surface at the two points is greater than the maximum height of the topography 
    262 at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ 
    263  
    264 \begin{equation} \label{dyn_ll_tmp2} 
    265 \begin{split} 
    266 \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\ 
    267 & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\ 
    268 & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) . 
    269 \end{split} 
    270 \end{equation}  
    271  
    272 If $\mathrm{ll\_tmp1}$ is true then the surface pressure gradient, zhpi at the $(i,j)$ 
    273 point is unmodified. If both logicals are false zhpi is set to zero. 
    274  
    275 If $\mathrm{ll\_tmp1}$ is true and $\mathrm{ll\_tmp2}$ is false then the surface pressure 
    276 gradient is multiplied through by zcpx which is the absolute value of the difference in 
    277 the water depths at the two points divided by the difference in the surface heights at the 
    278 two points. Thus the sign of the sea surface height gradient is retained but the magnitude 
    279 of the pressure force is determined by the difference in water depths rather than the 
    280 difference in surface height between the two points. Note that dividing by the difference 
    281 between the sea surface heights can be problematic if the heights approach parity. An 
    282 additional condition is applied to $\mathrm{ ll\_tmp2 }$ to ensure it is .false. in such 
    283 conditions. 
    284  
    285 \subsection   [Additional considerations (\textit{usrdef\_zgr})] 
    286          {Additional considerations (\mdl{usrdef\_zgr})} 
    287 \label{WAD_additional} 
    288  
    289 In the very shallow water where wetting and drying occurs the parametrisation of  
    290 bottom drag is clearly very important. In order to promote stability   
    291 it is sometimes useful to calculate the bottom drag using an implicit time-stepping approach.   
    292  
    293 Suitable specifcation of the surface heat flux in wetting and drying domains in forced and  
    294 coupled simulations needs further consideration. In order to prevent freezing or boiling 
    295 in uncoupled integrations the net surface heat fluxes need to be appropriately limited.   
    29627  
    29728%---------------------------------------------------------------------------------------- 
    29829%      The WAD test cases 
    29930%---------------------------------------------------------------------------------------- 
    300 \subsection   [The WAD test cases (\textit{usrdef\_zgr})] 
     31\section   [The WAD test cases (\textit{usrdef\_zgr})] 
    30132         {The WAD test cases (\mdl{usrdef\_zgr})} 
    30233\label{WAD_test_cases} 
    … …  
    32758 
    32859\clearpage 
    329 \subsubsection [WAD test case 1 : A simple linear slope] 
     60\subsection [WAD test case 1 : A simple linear slope] 
    33061                    {WAD test case 1 : A simple linear slope} 
    33162\label{WAD_test_case1} 
    … …  
    35182 
    35283\clearpage 
    353 \subsubsection [WAD test case 2 : A parabolic channel ] 
     84\subsection [WAD test case 2 : A parabolic channel ] 
    35485                    {WAD test case 2 : A parabolic channel} 
    35586\label{WAD_test_case2} 
    … …  
    374105 
    375106\clearpage 
    376 \subsubsection [WAD test case 3 : A parabolic channel (extreme slope) ] 
     107\subsection [WAD test case 3 : A parabolic channel (extreme slope) ] 
    377108                    {WAD test case 3 : A parabolic channel (extreme slope)} 
    378109\label{WAD_test_case3} 
    … …  
    393124 
    394125\clearpage 
    395 \subsubsection [WAD test case 4 : A parabolic bowl ] 
     126\subsection [WAD test case 4 : A parabolic bowl ] 
    396127                    {WAD test case 4 : A parabolic bowl} 
    397128\label{WAD_test_case4} 
    … …  
    415146 
    416147\clearpage 
    417 \subsubsection [WAD test case 5 : A double slope with shelf channel ] 
     148\subsection [WAD test case 5 : A double slope with shelf channel ] 
    418149                    {WAD test case 5 : A double slope with shelf channel} 
    419150\label{WAD_test_case5} 
    … …  
    434165 
    435166\clearpage 
    436 \subsubsection [WAD test case 6 : A parabolic channel with central bar ] 
     167\subsection [WAD test case 6 : A parabolic channel with central bar ] 
    437168                    {WAD test case 6 : A parabolic channel with central bar} 
    438169\label{WAD_test_case6} 
    … …  
    463194 
    464195\clearpage 
    465 \subsubsection [WAD test case 7 : A double slope with shelf, open-ended channel ] 
     196\subsection [WAD test case 7 : A double slope with shelf, open-ended channel ] 
    466197                    {WAD test case 7 : A double slope with shelf, open-ended channel} 
    467198\label{WAD_test_case7} 
    … …  
    511242% ================================================================ 
    512243 
    513 \bibliographystyle{wileyqj} 
    514 \bibliography{references} 
     244%\bibliographystyle{wileyqj} 
     245%\bibliographystyle{../../../doc/latex/NEMO/main/ametsoc.bst} 
     246%\bibliography{references} 
    515247 
    516248\end{document} 
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