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Changeset 10472 for NEMO/branches/UKMO/dev_10448_WAD_SBC_BUGFIX/doc/latex/NEMO/subfiles/chap_DYN.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/doc/latex/NEMO/subfiles/chap_DYN.tex

    r10442 r10472  
    13231323 
    13241324% ================================================================ 
     1325% Wetting and drying  
     1326% ================================================================ 
     1327\section{Wetting and drying } 
     1328\label{sec:DYN_wetdry} 
     1329There are two main options for wetting and drying code (wd): 
     1330(a) an iterative limiter (il) and (b) a directional limiter (dl). 
     1331The directional limiter is based on the scheme developed by \cite{WarnerEtal13} for RO 
     1332MS 
     1333which was in turn based on ideas developed for POM by \cite{Oey06}. The iterative 
     1334limiter is a new scheme.  The iterative limiter is activated by setting $\mathrm{ln\_wd\_il} = \mathrm{.true.}$ 
     1335and $\mathrm{ln\_wd\_dl} = \mathrm{.false.}$. The directional limiter is activated 
     1336by setting $\mathrm{ln\_wd\_dl} = \mathrm{.true.}$ and $\mathrm{ln\_wd\_il} = \mathrm{.false.}$. 
     1337 
     1338\nlst{namwad} 
     1339 
     1340The following terminology is used. The depth of the topography (positive downwards) 
     1341at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the NEMO code. 
     1342The height of the free surface (positive upwards) is denoted by $ \mathrm{ssh}$. Given the sign 
     1343conventions used, the water depth, $h$, is the height of the free surface plus the depth of the 
     1344topography (i.e. $\mathrm{ssh} + \mathrm{ht\_wd}$). 
     1345 
     1346Both wd schemes take all points in the domain below a land elevation of $\mathrm{rn\_wdld}$ to be 
     1347covered by water. They require the topography specified with a model 
     1348configuration to have negative depths at points where the land is higher than the 
     1349topography's reference sea-level. The vertical grid in NEMO is normally computed relative to an 
     1350initial state with zero sea surface height elevation. 
     1351The user can choose to compute the vertical grid and heights in the model relative to 
     1352a non-zero reference height for the free surface. This choice affects the calculation of the metrics and depths 
     1353(i.e. the $\mathrm{e3t\_0, ht\_0}$ etc. arrays). 
     1354 
     1355Points where the water depth is less than $\mathrm{rn\_wdmin1}$ are interpreted as ``dry''. 
     1356$\mathrm{rn\_wdmin1}$ is usually chosen to be of order $0.05$m but extreme topographies 
     1357with very steep slopes require larger values for normal choices of time-step. Surface fluxes 
     1358are also switched off for dry cells to prevent freezing, boiling etc. of very thin water layers. 
     1359The fluxes are tappered down using a $\mathrm{tanh}$ weighting function 
     1360to no flux as the dry limit $\mathrm{rn\_wdmin1}$ is approached. Even wet cells can be very shallow. 
     1361The depth at which to start tapering is controlled by the user by setting $\mathrm{rn\_wd\_sbcdep}$. 
     1362The fraction $(<1)$ of sufrace fluxes to use at this depth is set by $\mathrm{rn\_wd\_sbcfra}$. 
     1363 
     1364Both versions of the code have been tested in six test cases provided in the WAD\_TEST\_CASES configuration 
     1365and in ``realistic'' configurations covering parts of the north-west European shelf. 
     1366All these configurations have used pure sigma coordinates. It is expected that 
     1367the wetting and drying code will work in domains with more general s-coordinates provided 
     1368the coordinates are pure sigma in the region where wetting and drying actually occurs.  
     1369 
     1370The next sub-section descrbies the directional limiter and the following sub-section the iterative limiter. 
     1371The final sub-section covers some additional considerations that are relevant to both schemes. 
     1372 
     1373 
     1374%----------------------------------------------------------------------------------------- 
     1375%   Iterative limiters 
     1376%----------------------------------------------------------------------------------------- 
     1377\subsection   [Directional limiter (\textit{wet\_dry})] 
     1378         {Directional limiter (\mdl{wet\_dry})} 
     1379\label{subsec:DYN_wd_directional_limiter} 
     1380The principal idea of the directional limiter is that 
     1381water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than rn\_wdmin1). 
     1382 
     1383All the changes associated with this option are made to the barotropic solver for the non-linear 
     1384free surface code within dynspg\_ts. 
     1385On each barotropic sub-step the scheme determines the direction of the flow across each face of all the tracer cells 
     1386and sets the flux across the face to zero when the flux is from a dry tracer cell. This prevents cells 
     1387whose depth is rn\_wdmin1 or less from drying out further. The scheme does not force $h$ (the water depth) at tracer cells 
     1388to be at least the minimum depth and hence is able to conserve mass / volume. 
     1389 
     1390The flux across each $u$-face of a tracer cell is multiplied by a factor zuwdmask (an array which depends on ji and jj). 
     1391If the user sets ln\_wd\_dl\_ramp = .False. then zuwdmask is 1 when the 
     1392flux is from a cell with water depth greater than rn\_wdmin1 and 0 otherwise. If the user sets 
     1393ln\_wd\_dl\_ramp = .True. the flux across the face is ramped down as the water depth decreases 
     1394from 2 * rn\_wdmin1 to rn\_wdmin1. The use of this ramp reduced grid-scale noise in idealised test cases. 
     1395 
     1396At the point where the flux across a $u$-face is multiplied by zuwdmask , we have chosen 
     1397also to multiply the corresponding velocity on the ``now'' step at that face by zuwdmask. We could have 
     1398chosen not to do that and to allow fairly large velocities to occur in these ``dry'' cells. 
     1399The rationale for setting the velocity to zero is that it is the momentum equations that are being solved 
     1400and the total momentum of the upstream cell (treating it as a finite volume) should be considered 
     1401to be its depth times its velocity. This depth is considered to be zero at ``dry'' $u$-points consistent with its 
     1402treatment in the calculation of the flux of mass across the cell face. 
     1403 
     1404 
     1405\cite{WarnerEtal13} state that in their scheme the velocity masks at the cell faces for the baroclinic 
     1406timesteps are set to 0 or 1 depending on whether the average of the masks over the barotropic sub-steps is respectively less than 
     1407or greater than 0.5. That scheme does not conserve tracers in integrations started from constant tracer 
     1408fields (tracers independent of $x$, $y$ and $z$). Our scheme conserves constant tracers because 
     1409the velocities used at the tracer cell faces on the baroclinic timesteps are carefully calculated by dynspg\_ts 
     1410to equal their mean value during the barotropic steps. If the user sets ln\_wd\_dl\_bc = .True., the 
     1411baroclinic velocities are also multiplied by a suitably weighted average of zuwdmask.   
     1412 
     1413%----------------------------------------------------------------------------------------- 
     1414%   Iterative limiters 
     1415%----------------------------------------------------------------------------------------- 
     1416 
     1417\subsection   [Iterative limiter (\textit{wet\_dry})] 
     1418         {Iterative limiter (\mdl{wet\_dry})} 
     1419\label{subsec:DYN_wd_iterative_limiter} 
     1420 
     1421\subsubsection [Iterative flux limiter (\textit{wet\_dry})] 
     1422         {Iterative flux limiter (\mdl{wet\_dry})} 
     1423\label{subsubsec:DYN_wd_il_spg_limiter} 
     1424 
     1425The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry'' 
     1426or may become dry within the next time-step using an iterative method. 
     1427 
     1428The flux limiter for the barotropic flow (devised by Hedong Liu) can be understood as follows: 
     1429 
     1430The continuity equation for the total water depth in a column 
     1431\begin{equation} \label{dyn_wd_continuity} 
     1432 \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 . 
     1433\end{equation} 
     1434can be written in discrete form  as 
     1435 
     1436\begin{align} \label{dyn_wd_continuity_2} 
     1437\frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) )  
     1438&= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j}  + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\ 
     1439&= \mathrm{zzflx}_{i,j} . 
     1440\end{align} 
     1441 
     1442In the above $h$ is the depth of the water in the column at point $(i,j)$, 
     1443$\mathrm{flxu}_{i+1,j}$ is the flux out of the ``eastern'' face of the cell and 
     1444$\mathrm{flxv}_{i,j+1}$ the flux out of the ``northern'' face of the cell; $t_{n+1}$ is 
     1445the new timestep, $t_e$ is the old timestep (either $t_b$ or $t_n$) and $ \Delta t = 
     1446t_{n+1} - t_e$; $e_1 e_2$ is the area of the tracer cells centred at $(i,j)$ and 
     1447$\mathrm{zzflx}$ is the sum of the fluxes through all the faces. 
     1448 
     1449The flux limiter splits the flux $\mathrm{zzflx}$ into fluxes that are out of the cell 
     1450(zzflxp) and fluxes that are into the cell (zzflxn).  Clearly 
     1451 
     1452\begin{equation} \label{dyn_wd_zzflx_p_n_1} 
     1453\mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} .   
     1454\end{equation} 
     1455 
     1456The flux limiter iteratively adjusts the fluxes $\mathrm{flxu}$ and $\mathrm{flxv}$ until 
     1457none of the cells will ``dry out''. To be precise the fluxes are limited until none of the 
     1458cells has water depth less than $\mathrm{rn\_wdmin1}$ on step $n+1$. 
     1459 
     1460Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and 
     1461$\mathrm{flxv}^{(m)}$.  Then the adjustment is achieved by seeking a set of coefficients, 
     1462$\mathrm{zcoef}_{i,j}^{(m)}$ such that: 
     1463 
     1464\begin{equation} \label{dyn_wd_continuity_coef} 
     1465\begin{split} 
     1466\mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\ 
     1467\mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j} 
     1468\end{split} 
     1469\end{equation} 
     1470 
     1471where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around 
     1472cells that would otherwise dry. 
     1473 
     1474The iteration is initialised by setting 
     1475 
     1476\begin{equation} \label{dyn_wd_zzflx_initial} 
     1477\mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad  \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} .  
     1478\end{equation} 
     1479 
     1480The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the 
     1481cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell 
     1482times the timestep divided by the cell area. Using (\ref{dyn_wd_continuity_2}) this 
     1483condition is 
     1484 
     1485\begin{equation} \label{dyn_wd_continuity_if} 
     1486h_{i,j}(t_e)  - \mathrm{rn\_wdmin1} <  \frac{\Delta t}{e_1 e_2} ( \mathrm{zzflxp}^{(m)}_{i,j} + \mathrm{zzflxn}^{(m)}_{i,j} ) . 
     1487\end{equation} 
     1488 
     1489Rearranging (\ref{dyn_wd_continuity_if}) we can obtain an expression for the maximum 
     1490outward flux that can be allowed and still maintain the minimum wet depth: 
     1491 
     1492\begin{equation} \label{dyn_wd_max_flux} 
     1493\begin{split} 
     1494\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{]} \\ 
     1495\phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] 
     1496\end{split} 
     1497\end{equation} 
     1498 
     1499Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\it [Q: Why is 
     1500this necessary/desirable?]}. Substituting from (\ref{dyn_wd_continuity_coef}) gives an 
     1501expression for the coefficient needed to multiply the outward flux at this cell in order 
     1502to avoid drying. 
     1503 
     1504\begin{equation} \label{dyn_wd_continuity_nxtcoef} 
     1505\begin{split} 
     1506\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{]} \\ 
     1507\phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} }  
     1508\end{split} 
     1509\end{equation} 
     1510 
     1511Only the outward flux components are altered but, of course, outward fluxes from one cell 
     1512are inward fluxes to adjacent cells and the balance in these cells may need subsequent 
     1513adjustment; hence the iterative nature of this scheme.  Note, for example, that the flux 
     1514across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration 
     1515if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then 
     1516the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by 
     1517the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across 
     1518the faces of the cells do not ``compete'' (they do not over-write each other) and one 
     1519would expect the scheme to converge relatively quickly. The scheme is flux based so 
     1520conserves mass. It also conserves constant tracers for the same reason that the 
     1521directional limiter does. 
     1522 
     1523 
     1524%---------------------------------------------------------------------------------------- 
     1525%      Surface pressure gradients 
     1526%---------------------------------------------------------------------------------------- 
     1527\subsubsection   [Modification of surface pressure gradients (\textit{dynhpg})] 
     1528         {Modification of surface pressure gradients (\mdl{dynhpg})} 
     1529\label{subsubsec:DYN_wd_il_spg} 
     1530 
     1531At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the 
     1532topography is sloping at these points the sea-surface will have a similar slope and there 
     1533will hence be very large horizontal pressure gradients at these points. The WAD modifies 
     1534the magnitude but not the sign of the surface pressure gradients (zhpi and zhpj) at such 
     1535points by mulitplying them by positive factors (zcpx and zcpy respectively) that lie 
     1536between $0$ and $1$. 
     1537 
     1538We describe how the scheme works for the ``eastward'' pressure gradient, zhpi, calculated 
     1539at the $(i,j)$th $u$-point. The scheme uses the ht\_wd depths and surface heights at the 
     1540neighbouring $(i+1,j)$ and $(i,j)$ tracer points.  zcpx is calculated using two logicals 
     1541variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid 
     1542column.  The three possible combinations are illustrated in figure \ref{Fig_WAD_dynhpg}. 
     1543 
     1544%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     1545\begin{figure}[!ht] \begin{center} 
     1546\includegraphics[width=0.8\textwidth]{Fig_WAD_dynhpg} 
     1547\caption{ \label{Fig_WAD_dynhpg} 
     1548Illustrations of the three possible combinations of the logical variables controlling the 
     1549limiting of the horizontal pressure gradient in wetting and drying regimes} 
     1550\end{center}\end{figure} 
     1551%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     1552 
     1553The first logical, $\mathrm{ll\_tmp1}$, is set to true if and only if the water depth at 
     1554both neighbouring points is greater than $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ and 
     1555the minimum height of the sea surface at the two points is greater than the maximum height 
     1556of the topography at the two points: 
     1557 
     1558\begin{equation} \label{dyn_ll_tmp1} 
     1559\begin{split} 
     1560\mathrm{ll\_tmp1}  = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\ 
     1561                     & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\  .and.} \\ 
     1562& \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\ 
     1563& \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\ 
     1564& \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 } 
     1565\end{split} 
     1566\end{equation} 
     1567 
     1568The second logical, $\mathrm{ll\_tmp2}$, is set to true if and only if the maximum height 
     1569of the sea surface at the two points is greater than the maximum height of the topography 
     1570at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ 
     1571 
     1572\begin{equation} \label{dyn_ll_tmp2} 
     1573\begin{split} 
     1574\mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\ 
     1575& \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\ 
     1576& \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) . 
     1577\end{split} 
     1578\end{equation} 
     1579 
     1580If $\mathrm{ll\_tmp1}$ is true then the surface pressure gradient, zhpi at the $(i,j)$ 
     1581point is unmodified. If both logicals are false zhpi is set to zero. 
     1582 
     1583If $\mathrm{ll\_tmp1}$ is true and $\mathrm{ll\_tmp2}$ is false then the surface pressure 
     1584gradient is multiplied through by zcpx which is the absolute value of the difference in 
     1585the water depths at the two points divided by the difference in the surface heights at the 
     1586two points. Thus the sign of the sea surface height gradient is retained but the magnitude 
     1587of the pressure force is determined by the difference in water depths rather than the 
     1588difference in surface height between the two points. Note that dividing by the difference 
     1589between the sea surface heights can be problematic if the heights approach parity. An 
     1590additional condition is applied to $\mathrm{ ll\_tmp2 }$ to ensure it is .false. in such 
     1591conditions. 
     1592 
     1593\subsubsection   [Additional considerations (\textit{usrdef\_zgr})] 
     1594         {Additional considerations (\mdl{usrdef\_zgr})} 
     1595\label{subsubsec:WAD_additional} 
     1596 
     1597In the very shallow water where wetting and drying occurs the parametrisation of 
     1598bottom drag is clearly very important. In order to promote stability 
     1599it is sometimes useful to calculate the bottom drag using an implicit time-stepping approach. 
     1600 
     1601Suitable specifcation of the surface heat flux in wetting and drying domains in forced and 
     1602coupled simulations needs further consideration. In order to prevent freezing or boiling 
     1603in uncoupled integrations the net surface heat fluxes need to be appropriately limited. 
     1604 
     1605%---------------------------------------------------------------------------------------- 
     1606%      The WAD test cases 
     1607%---------------------------------------------------------------------------------------- 
     1608\subsection   [The WAD test cases (\textit{usrdef\_zgr})] 
     1609         {The WAD test cases (\mdl{usrdef\_zgr})} 
     1610\label{WAD_test_cases} 
     1611 
     1612See the WAD tests MY\_DOC documention for details of the WAD test cases. 
     1613 
     1614 
     1615 
     1616% ================================================================ 
    13251617% Time evolution term  
    13261618% ================================================================ 
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