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Changeset 10472 for NEMO/branches/UKMO/dev_10448_WAD_SBC_BUGFIX/tests/WAD/MY_DOCS/WAD_doc.tex – NEMO

Ignore:
Timestamp:
2019-01-08T17:17:38+01:00 (5 years ago)
Author:
deazer
Message:

Split WAD documentation into NEMO book and Test cases only

File:
1 edited

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  • NEMO/branches/UKMO/dev_10448_WAD_SBC_BUGFIX/tests/WAD/MY_DOCS/WAD_doc.tex

    r10467 r10472  
    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. Surface fluxes  
    58 are also switched off for dry cells to prevent freezing, boiling etc. of very thin water layers. 
    59 The fluxes are tappered down using a $\mathrm{tanh}$ weighting function  
    60 to no flux as the dry limit $\mathrm{rn\_wdmin1}$ is approached. Even wet cells can be very shallow. 
    61 The depth at which to start tapering is controlled by the user by setting $\mathrm{rn\_wd\_sbcdep}$. 
    62 The fraction $(<1)$ of sufrace fluxes to use at this depth is set by $\mathrm{rn\_wd\_sbcfra}$. 
    63  
    64 Both versions of the code have been tested in six test cases provided in the WAD\_TEST\_CASES configuration 
    65 and in ``realistic'' configurations covering parts of the north-west European shelf.  
    66 All these configurations have used pure sigma coordinates. It is expected that 
    67 the wetting and drying code will work in domains with more general s-coordinates provided 
    68 the coordinates are pure sigma in the region where wetting and drying actually occurs.   
    69  
    70 The next sub-section descrbies the directional limiter and the following sub-section the iterative limiter.  
    71 The final sub-section covers some additional considerations that are relevant to both schemes.  
    72  
    73 %----------------------------------------------------------------------------------------- 
    74 %   Iterative limiters 
    75 %----------------------------------------------------------------------------------------- 
    76 \subsection   [Directional limiter (\textit{wet\_dry})] 
    77          {Directional limiter (\mdl{wet\_dry})} 
    78 \label{DYN_wd_directional_limiter} 
    79  
    80 The principal idea of the directional limiter is that  
    81 water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than rn\_wdmin1). 
    82  
    83 All the changes associated with this option are made to the barotropic solver for the non-linear  
    84 free surface code within dynspg\_ts.  
    85 On each barotropic sub-step the scheme determines the direction of the flow across each face of all the tracer cells 
    86 and sets the flux across the face to zero when the flux is from a dry tracer cell. This prevents cells 
    87 whose depth is rn\_wdmin1 or less from drying out further. The scheme does not force $h$ (the water depth) at tracer cells 
    88 to be at least the minimum depth and hence is able to conserve mass / volume.  
    89  
    90 The flux across each $u$-face of a tracer cell is multiplied by a factor zuwdmask (an array which depends on ji and jj).  
    91 If the user sets ln\_wd\_dl\_ramp = .False. then zuwdmask is 1 when the 
    92 flux is from a cell with water depth greater than rn\_wdmin1 and 0 otherwise. If the user sets 
    93 ln\_wd\_dl\_ramp = .True. the flux across the face is ramped down as the water depth decreases 
    94 from 2 * rn\_wdmin1 to rn\_wdmin1. The use of this ramp reduced grid-scale noise in idealised test cases.  
    95  
    96 At the point where the flux across a $u$-face is multiplied by zuwdmask , we have chosen 
    97 also to multiply the corresponding velocity on the ``now'' step at that face by zuwdmask. We could have 
    98 chosen not to do that and to allow fairly large velocities to occur in these ``dry'' cells.  
    99 The rationale for setting the velocity to zero is that it is the momentum equations that are being solved 
    100 and the total momentum of the upstream cell (treating it as a finite volume) should be considered 
    101 to be its depth times its velocity. This depth is considered to be zero at ``dry'' $u$-points consistent with its  
    102 treatment in the calculation of the flux of mass across the cell face.          
    103  
    104 \cite{WarnerEtal13} state that in their scheme the velocity masks at the cell faces for the baroclinic  
    105 timesteps are set to 0 or 1 depending on whether the average of the masks over the barotropic sub-steps is respectively less than  
    106 or greater than 0.5. That scheme does not conserve tracers in integrations started from constant tracer 
    107 fields (tracers independent of $x$, $y$ and $z$). Our scheme conserves constant tracers because 
    108 the velocities used at the tracer cell faces on the baroclinic timesteps are carefully calculated by dynspg\_ts 
    109 to equal their mean value during the barotropic steps. If the user sets ln\_wd\_dl\_bc = .True., the 
    110 baroclinic velocities are also multiplied by a suitably weighted average of zuwdmask.       
    111       
    112 %----------------------------------------------------------------------------------------- 
    113 %   Iterative limiters 
    114 %----------------------------------------------------------------------------------------- 
    115 \subsection   [Iterative limiter (\textit{wet\_dry})] 
    116          {Iterative limiter (\mdl{wet\_dry})} 
    117 \label{DYN_wd_iterative_limiter} 
    118  
    119 \subsubsection [Iterative flux limiter (\textit{wet\_dry})] 
    120          {Iterative flux limiter (\mdl{wet\_dry})} 
    121 \label{DYN_wd_il_spg_limiter} 
    122  
    123 The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry'' 
    124 or may become dry within the next time-step using an iterative method.  
    125  
    126 The flux limiter for the barotropic flow (devised by Hedong Liu) can be understood as follows:  
    127  
    128 The continuity equation for the total water depth in a column  
    129 \begin{equation} \label{dyn_wd_continuity} 
    130  \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 . 
    131 \end{equation}  
    132 can be written in discrete form  as   
    133  
    134 \begin{align} \label{dyn_wd_continuity_2} 
    135 \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) )  
    136 &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j}  + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\ 
    137 &= \mathrm{zzflx}_{i,j} . 
    138 \end{align}  
    139  
    140 In the above $h$ is the depth of the water in the column at point $(i,j)$, 
    141 $\mathrm{flxu}_{i+1,j}$ is the flux out of the ``eastern'' face of the cell and 
    142 $\mathrm{flxv}_{i,j+1}$ the flux out of the ``northern'' face of the cell; $t_{n+1}$ is 
    143 the new timestep, $t_e$ is the old timestep (either $t_b$ or $t_n$) and $ \Delta t = 
    144 t_{n+1} - t_e$; $e_1 e_2$ is the area of the tracer cells centred at $(i,j)$ and 
    145 $\mathrm{zzflx}$ is the sum of the fluxes through all the faces. 
    146  
    147 The flux limiter splits the flux $\mathrm{zzflx}$ into fluxes that are out of the cell 
    148 (zzflxp) and fluxes that are into the cell (zzflxn).  Clearly 
    149  
    150 \begin{equation} \label{dyn_wd_zzflx_p_n_1} 
    151 \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} .   
    152 \end{equation}  
    153  
    154 The flux limiter iteratively adjusts the fluxes $\mathrm{flxu}$ and $\mathrm{flxv}$ until 
    155 none of the cells will ``dry out''. To be precise the fluxes are limited until none of the 
    156 cells has water depth less than $\mathrm{rn\_wdmin1}$ on step $n+1$. 
    157  
    158 Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and 
    159 $\mathrm{flxv}^{(m)}$.  Then the adjustment is achieved by seeking a set of coefficients, 
    160 $\mathrm{zcoef}_{i,j}^{(m)}$ such that: 
    161  
    162 \begin{equation} \label{dyn_wd_continuity_coef} 
    163 \begin{split} 
    164 \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\ 
    165 \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j} 
    166 \end{split} 
    167 \end{equation}  
    168   
    169 where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around 
    170 cells that would otherwise dry. 
    171  
    172 The iteration is initialised by setting 
    173  
    174 \begin{equation} \label{dyn_wd_zzflx_initial} 
    175 \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad  \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} .  
    176 \end{equation}  
    177  
    178 The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the 
    179 cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell 
    180 times the timestep divided by the cell area. Using (\ref{dyn_wd_continuity_2}) this 
    181 condition is 
    182  
    183 \begin{equation} \label{dyn_wd_continuity_if} 
    184 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} ) . 
    185 \end{equation}  
    186  
    187 Rearranging (\ref{dyn_wd_continuity_if}) we can obtain an expression for the maximum 
    188 outward flux that can be allowed and still maintain the minimum wet depth: 
    189  
    190 \begin{equation} \label{dyn_wd_max_flux} 
    191 \begin{split} 
    192 \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{]} \\ 
    193 \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] 
    194 \end{split} 
    195 \end{equation} 
    196  
    197 Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\it [Q: Why is 
    198 this necessary/desirable?]}. Substituting from (\ref{dyn_wd_continuity_coef}) gives an 
    199 expression for the coefficient needed to multiply the outward flux at this cell in order 
    200 to avoid drying.  
    201  
    202 \begin{equation} \label{dyn_wd_continuity_nxtcoef} 
    203 \begin{split} 
    204 \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{]} \\ 
    205 \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} }  
    206 \end{split} 
    207 \end{equation}  
    208  
    209 Only the outward flux components are altered but, of course, outward fluxes from one cell 
    210 are inward fluxes to adjacent cells and the balance in these cells may need subsequent 
    211 adjustment; hence the iterative nature of this scheme.  Note, for example, that the flux 
    212 across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration 
    213 if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then 
    214 the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by 
    215 the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across 
    216 the faces of the cells do not ``compete'' (they do not over-write each other) and one 
    217 would expect the scheme to converge relatively quickly. The scheme is flux based so 
    218 conserves mass. It also conserves constant tracers for the same reason that the  
    219 directional limiter does.   
    220  
    221  
    222 %---------------------------------------------------------------------------------------- 
    223 %      Surface pressure gradients 
    224 %---------------------------------------------------------------------------------------- 
    225 \subsubsection   [Modification of surface pressure gradients (\textit{dynhpg})] 
    226          {Modification of surface pressure gradients (\mdl{dynhpg})} 
    227 \label{DYN_wd_il_spg} 
    228  
    229 At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the 
    230 topography is sloping at these points the sea-surface will have a similar slope and there 
    231 will hence be very large horizontal pressure gradients at these points. The WAD modifies 
    232 the magnitude but not the sign of the surface pressure gradients (zhpi and zhpj) at such 
    233 points by mulitplying them by positive factors (zcpx and zcpy respectively) that lie 
    234 between $0$ and $1$. 
    235  
    236 We describe how the scheme works for the ``eastward'' pressure gradient, zhpi, calculated 
    237 at the $(i,j)$th $u$-point. The scheme uses the ht\_wd depths and surface heights at the 
    238 neighbouring $(i+1,j)$ and $(i,j)$ tracer points.  zcpx is calculated using two logicals 
    239 variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid 
    240 column.  The three possible combinations are illustrated in figure \ref{Fig_WAD_dynhpg}. 
    241 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    242 \begin{figure}[!ht] \begin{center} 
    243 \includegraphics[width=0.8\textwidth]{Fig_WAD_dynhpg} 
    244 \caption{ \label{Fig_WAD_dynhpg} 
    245 Illustrations of the three possible combinations of the logical variables controlling the  
    246 limiting of the horizontal pressure gradient in wetting and drying regimes} 
    247 \end{center}\end{figure} 
    248 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    249  
    250 The first logical, $\mathrm{ll\_tmp1}$, is set to true if and only if the water depth at 
    251 both neighbouring points is greater than $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ and 
    252 the minimum height of the sea surface at the two points is greater than the maximum height 
    253 of the topography at the two points: 
    254  
    255 \begin{equation} \label{dyn_ll_tmp1} 
    256 \begin{split} 
    257 \mathrm{ll\_tmp1}  = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\ 
    258                      & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\  .and.} \\ 
    259 & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\ 
    260 & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\ 
    261 & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 } 
    262 \end{split} 
    263 \end{equation}  
    264  
    265 The second logical, $\mathrm{ll\_tmp2}$, is set to true if and only if the maximum height 
    266 of the sea surface at the two points is greater than the maximum height of the topography 
    267 at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ 
    268  
    269 \begin{equation} \label{dyn_ll_tmp2} 
    270 \begin{split} 
    271 \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\ 
    272 & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\ 
    273 & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) . 
    274 \end{split} 
    275 \end{equation}  
    276  
    277 If $\mathrm{ll\_tmp1}$ is true then the surface pressure gradient, zhpi at the $(i,j)$ 
    278 point is unmodified. If both logicals are false zhpi is set to zero. 
    279  
    280 If $\mathrm{ll\_tmp1}$ is true and $\mathrm{ll\_tmp2}$ is false then the surface pressure 
    281 gradient is multiplied through by zcpx which is the absolute value of the difference in 
    282 the water depths at the two points divided by the difference in the surface heights at the 
    283 two points. Thus the sign of the sea surface height gradient is retained but the magnitude 
    284 of the pressure force is determined by the difference in water depths rather than the 
    285 difference in surface height between the two points. Note that dividing by the difference 
    286 between the sea surface heights can be problematic if the heights approach parity. An 
    287 additional condition is applied to $\mathrm{ ll\_tmp2 }$ to ensure it is .false. in such 
    288 conditions. 
    289  
    290 \subsection   [Additional considerations (\textit{usrdef\_zgr})] 
    291          {Additional considerations (\mdl{usrdef\_zgr})} 
    292 \label{WAD_additional} 
    293  
    294 In the very shallow water where wetting and drying occurs the parametrisation of  
    295 bottom drag is clearly very important. In order to promote stability   
    296 it is sometimes useful to calculate the bottom drag using an implicit time-stepping approach.   
    297  
    298 Suitable specifcation of the surface heat flux in wetting and drying domains in forced and  
    299 coupled simulations needs further consideration. In order to prevent freezing or boiling 
    300 in uncoupled integrations the net surface heat fluxes need to be appropriately limited.   
    30127  
    30228%---------------------------------------------------------------------------------------- 
    30329%      The WAD test cases 
    30430%---------------------------------------------------------------------------------------- 
    305 \subsection   [The WAD test cases (\textit{usrdef\_zgr})] 
     31\section   [The WAD test cases (\textit{usrdef\_zgr})] 
    30632         {The WAD test cases (\mdl{usrdef\_zgr})} 
    30733\label{WAD_test_cases} 
     
    33258 
    33359\clearpage 
    334 \subsubsection [WAD test case 1 : A simple linear slope] 
     60\subsection [WAD test case 1 : A simple linear slope] 
    33561                    {WAD test case 1 : A simple linear slope} 
    33662\label{WAD_test_case1} 
     
    35682 
    35783\clearpage 
    358 \subsubsection [WAD test case 2 : A parabolic channel ] 
     84\subsection [WAD test case 2 : A parabolic channel ] 
    35985                    {WAD test case 2 : A parabolic channel} 
    36086\label{WAD_test_case2} 
     
    379105 
    380106\clearpage 
    381 \subsubsection [WAD test case 3 : A parabolic channel (extreme slope) ] 
     107\subsection [WAD test case 3 : A parabolic channel (extreme slope) ] 
    382108                    {WAD test case 3 : A parabolic channel (extreme slope)} 
    383109\label{WAD_test_case3} 
     
    398124 
    399125\clearpage 
    400 \subsubsection [WAD test case 4 : A parabolic bowl ] 
     126\subsection [WAD test case 4 : A parabolic bowl ] 
    401127                    {WAD test case 4 : A parabolic bowl} 
    402128\label{WAD_test_case4} 
     
    420146 
    421147\clearpage 
    422 \subsubsection [WAD test case 5 : A double slope with shelf channel ] 
     148\subsection [WAD test case 5 : A double slope with shelf channel ] 
    423149                    {WAD test case 5 : A double slope with shelf channel} 
    424150\label{WAD_test_case5} 
     
    439165 
    440166\clearpage 
    441 \subsubsection [WAD test case 6 : A parabolic channel with central bar ] 
     167\subsection [WAD test case 6 : A parabolic channel with central bar ] 
    442168                    {WAD test case 6 : A parabolic channel with central bar} 
    443169\label{WAD_test_case6} 
     
    468194 
    469195\clearpage 
    470 \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 ] 
    471197                    {WAD test case 7 : A double slope with shelf, open-ended channel} 
    472198\label{WAD_test_case7} 
     
    517243 
    518244%\bibliographystyle{wileyqj} 
    519 \bibliographystyle{../../../doc/latex/NEMO/main/ametsoc.bst} 
    520 \bibliography{references} 
     245%\bibliographystyle{../../../doc/latex/NEMO/main/ametsoc.bst} 
     246%\bibliography{references} 
    521247 
    522248\end{document} 
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