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1\include{Preamble}
2
3\begin{document}
4
5\title{Draft description of NEMO wetting and drying scheme:     29 November 2017 }
6
7\author{ Enda O'Dea, Hedong Liu, Jason Holt, Andrew Coward  and Michael J. Bell  }
8
9%------------------------------------------------------------------------
10% End of temporary latex header (to be removed)
11%------------------------------------------------------------------------
12
13% ================================================================
14% Chapter Ocean Dynamics (DYN)
15% ================================================================
16\chapter{Ocean Dynamics (DYN)}
17\label{DYN}
18\minitoc
19
20% add a figure for  dynvor ens, ene latices
21
22$\ $\newline    % force a new ligne
23
24% ================================================================
25% Wetting and drying
26% ================================================================
27\section{Wetting and drying }
28\label{DYN_wetdry}
29
30There are two main options for wetting and drying code (wd):
31(a) an iterative limiter (il) and (b) a directional limiter (dl).
32The directional limiter is based on the scheme developed by \cite{WarnerEtal13} for ROMS
33which was in turn based on ideas developed for POM by \cite{Oey06}. The iterative
34limiter is a new scheme.  The iterative limiter is activated by setting $\mathrm{ln\_wd\_il} = \mathrm{.true.}$
35and $\mathrm{ln\_wd\_dl} = \mathrm{.false.}$. The directional limiter is activated
36by setting $\mathrm{ln\_wd\_dl} = \mathrm{.true.}$ and $\mathrm{ln\_wd\_il} = \mathrm{.false.}$.
37
38\namdisplay{nam_wad}
39
40The following terminology is used. The depth of the topography (positive downwards)
41at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the NEMO code.
42The height of the free surface (positive upwards) is denoted by $ \mathrm{ssh}$. Given the sign
43conventions used, the water depth, $h$, is the height of the free surface plus the depth of the
44topography (i.e. $\mathrm{ssh} + \mathrm{ht\_wd}$).
45
46Both wd schemes take all points in the domain below a land elevation of $\mathrm{rn\_wdld}$ to be
47covered by water. They require the topography specified with a model
48configuration to have negative depths at points where the land is higher than the
49topography's reference sea-level. The vertical grid in NEMO is normally computed relative to an
50initial state with zero sea surface height elevation.
51The user can choose to compute the vertical grid and heights in the model relative to
52a 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
55Points 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
57with very steep slopes require larger values for normal choices of time-step.
58
59Both versions of the code have been tested in six test cases provided in the WAD\_TEST\_CASES configuration
60and in ``realistic'' configurations covering parts of the north-west European shelf.
61All these configurations have used pure sigma coordinates. It is expected that
62the wetting and drying code will work in domains with more general s-coordinates provided
63the coordinates are pure sigma in the region where wetting and drying actually occurs. 
64
65The next sub-section descrbies the directional limiter and the following sub-section the iterative limiter.
66The 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
75The principal idea of the directional limiter is that
76water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than rn\_wdmin1).
77
78All the changes associated with this option are made to the barotropic solver for the non-linear
79free surface code within dynspg\_ts.
80On each barotropic sub-step the scheme determines the direction of the flow across each face of all the tracer cells
81and sets the flux across the face to zero when the flux is from a dry tracer cell. This prevents cells
82whose depth is rn\_wdmin1 or less from drying out further. The scheme does not force $h$ (the water depth) at tracer cells
83to be at least the minimum depth and hence is able to conserve mass / volume.
84
85The flux across each $u$-face of a tracer cell is multiplied by a factor zuwdmask (an array which depends on ji and jj).
86If the user sets ln\_wd\_dl\_ramp = .False. then zuwdmask is 1 when the
87flux is from a cell with water depth greater than rn\_wdmin1 and 0 otherwise. If the user sets
88ln\_wd\_dl\_ramp = .True. the flux across the face is ramped down as the water depth decreases
89from 2 * rn\_wdmin1 to rn\_wdmin1. The use of this ramp reduced grid-scale noise in idealised test cases.
90
91At the point where the flux across a $u$-face is multiplied by zuwdmask , we have chosen
92also to multiply the corresponding velocity on the ``now'' step at that face by zuwdmask. We could have
93chosen not to do that and to allow fairly large velocities to occur in these ``dry'' cells.
94The rationale for setting the velocity to zero is that it is the momentum equations that are being solved
95and the total momentum of the upstream cell (treating it as a finite volume) should be considered
96to be its depth times its velocity. This depth is considered to be zero at ``dry'' $u$-points consistent with its
97treatment 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
100timesteps are set to 0 or 1 depending on whether the average of the masks over the barotropic sub-steps is respectively less than
101or greater than 0.5. That scheme does not conserve tracers in integrations started from constant tracer
102fields (tracers independent of $x$, $y$ and $z$). Our scheme conserves constant tracers because
103the velocities used at the tracer cell faces on the baroclinic timesteps are carefully calculated by dynspg\_ts
104to equal their mean value during the barotropic steps. If the user sets ln\_wd\_dl\_bc = .True., the
105baroclinic 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
118The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry''
119or may become dry within the next time-step using an iterative method.
120
121The flux limiter for the barotropic flow (devised by Hedong Liu) can be understood as follows:
122
123The 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} 
127can 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
135In 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
138the new timestep, $t_e$ is the old timestep (either $t_b$ or $t_n$) and $ \Delta t =
139t_{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
142The 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
149The flux limiter iteratively adjusts the fluxes $\mathrm{flxu}$ and $\mathrm{flxv}$ until
150none of the cells will ``dry out''. To be precise the fluxes are limited until none of the
151cells has water depth less than $\mathrm{rn\_wdmin1}$ on step $n+1$.
152
153Let 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 
164where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around
165cells that would otherwise dry.
166
167The 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
173The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the
174cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell
175times the timestep divided by the cell area. Using (\ref{dyn_wd_continuity_2}) this
176condition is
177
178\begin{equation} \label{dyn_wd_continuity_if}
179h_{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
182Rearranging (\ref{dyn_wd_continuity_if}) we can obtain an expression for the maximum
183outward 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
192Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\it [Q: Why is
193this necessary/desirable?]}. Substituting from (\ref{dyn_wd_continuity_coef}) gives an
194expression for the coefficient needed to multiply the outward flux at this cell in order
195to 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
204Only the outward flux components are altered but, of course, outward fluxes from one cell
205are inward fluxes to adjacent cells and the balance in these cells may need subsequent
206adjustment; hence the iterative nature of this scheme.  Note, for example, that the flux
207across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration
208if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then
209the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by
210the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across
211the faces of the cells do not ``compete'' (they do not over-write each other) and one
212would expect the scheme to converge relatively quickly. The scheme is flux based so
213conserves mass. It also conserves constant tracers for the same reason that the
214directional 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
224At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the
225topography is sloping at these points the sea-surface will have a similar slope and there
226will hence be very large horizontal pressure gradients at these points. The WAD modifies
227the magnitude but not the sign of the surface pressure gradients (zhpi and zhpj) at such
228points by mulitplying them by positive factors (zcpx and zcpy respectively) that lie
229between $0$ and $1$.
230
231We describe how the scheme works for the ``eastward'' pressure gradient, zhpi, calculated
232at the $(i,j)$th $u$-point. The scheme uses the ht\_wd depths and surface heights at the
233neighbouring $(i+1,j)$ and $(i,j)$ tracer points.  zcpx is calculated using two logicals
234variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid
235column.  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}
240Illustrations of the three possible combinations of the logical variables controlling the
241limiting of the horizontal pressure gradient in wetting and drying regimes}
242\end{center}\end{figure}
243%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
244
245The first logical, $\mathrm{ll\_tmp1}$, is set to true if and only if the water depth at
246both neighbouring points is greater than $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ and
247the minimum height of the sea surface at the two points is greater than the maximum height
248of 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
260The second logical, $\mathrm{ll\_tmp2}$, is set to true if and only if the maximum height
261of the sea surface at the two points is greater than the maximum height of the topography
262at 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
272If $\mathrm{ll\_tmp1}$ is true then the surface pressure gradient, zhpi at the $(i,j)$
273point is unmodified. If both logicals are false zhpi is set to zero.
274
275If $\mathrm{ll\_tmp1}$ is true and $\mathrm{ll\_tmp2}$ is false then the surface pressure
276gradient is multiplied through by zcpx which is the absolute value of the difference in
277the water depths at the two points divided by the difference in the surface heights at the
278two points. Thus the sign of the sea surface height gradient is retained but the magnitude
279of the pressure force is determined by the difference in water depths rather than the
280difference in surface height between the two points. Note that dividing by the difference
281between the sea surface heights can be problematic if the heights approach parity. An
282additional condition is applied to $\mathrm{ ll\_tmp2 }$ to ensure it is .false. in such
283conditions.
284
285\subsection   [Additional considerations (\textit{usrdef\_zgr})]
286         {Additional considerations (\mdl{usrdef\_zgr})}
287\label{WAD_additional}
288
289In the very shallow water where wetting and drying occurs the parametrisation of
290bottom drag is clearly very important. In order to promote stability 
291it is sometimes useful to calculate the bottom drag using an implicit time-stepping approach. 
292
293Suitable specifcation of the surface heat flux in wetting and drying domains in forced and
294coupled simulations needs further consideration. In order to prevent freezing or boiling
295in uncoupled integrations the net surface heat fluxes need to be appropriately limited. 
296 
297%----------------------------------------------------------------------------------------
298%      The WAD test cases
299%----------------------------------------------------------------------------------------
300\subsection   [The WAD test cases (\textit{usrdef\_zgr})]
301         {The WAD test cases (\mdl{usrdef\_zgr})}
302\label{WAD_test_cases}
303
304This section contains details of the seven test cases that can be run as part of the
305WAD\_TEST\_CASES configuration. All the test cases are shallow (less than 10m deep),
306basins or channels with 4m high walls and some of topography that can wet and dry up to
3072.5m above sea-level. The horizontal grid is uniform with a 1km resolution and measures
30852km by 34km. These dimensions are determined by a combination of code in the
309\mdl{usrdef\_nam} module located in the WAD\_TEST\_CASES/MY\_SRC directory and setting
310read in from the namusr\_def namelist. The first six test cases are closed systems with no
311rotation or external forcing and motion is simply initiated by an initial ssh slope. The
312seventh test case introduces and open boundary at the right-hand end of the channel which
313is forced with sinousoidally varying ssh and barotropic velocities.
314
315\namdisplay{nam_wad_usr}
316
317The $\mathrm{nn\_wad\_test}$ parameter can takes values 1 to 7 and it is this parameter
318that determines which of the test cases will be run. Most cases can be run with the
319default settings but the simple linear slope cases (tests 1 and 5) can be run with lower
320values of $\mathrm{rn\_wdmin1}$. Any recommended changes to the default namelist settings
321will be stated in the individual subsections.
322
323Test case 7 requires additional {\tt namelist\_cfg} changes to activate the open boundary
324and lengthen the duration of the run (in order to demonstrate the full forcing cycle).
325There is also a simple python script which needs to be run in order to generate the
326boundary forcing files.  Full details are given in subsection (\ref{WAD_test_case7}).
327
328\clearpage
329\subsubsection [WAD test case 1 : A simple linear slope]
330                    {WAD test case 1 : A simple linear slope}
331\label{WAD_test_case1}
332
333The first test case is a simple linear slope (in the x-direction, uniform in y) with an
334adverse SSH gradient that, when released, creates a surge up the slope. The parameters are
335chosen such that the surge rises above sea-level before falling back and oscillating
336towards an equilibrium position. This case can be run with $\mathrm{rn\_wdmin1}$ values as
337low as 0.075m. I.e. the following change may be made to the default values in {\tt
338namelist\_cfg} (for this test only):
339
340\namdisplay{nam_wad_tc1}
341
342%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
343\begin{figure}[htb] \begin{center}
344\includegraphics[width=0.8\textwidth]{Fig_WAD_TC1}
345\caption{ \label{Fig_WAD_TC1}
346The evolution of the sea surface height in WAD\_TEST\_CASE 1 from the initial state (t=0)
347over the first three hours of simulation. Note that in this time-frame the resultant surge
348reaches to nearly 2m above sea-level before retreating.}
349\end{center}\end{figure}
350%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
351
352\clearpage
353\subsubsection [WAD test case 2 : A parabolic channel ]
354                    {WAD test case 2 : A parabolic channel}
355\label{WAD_test_case2}
356
357The second and third test cases use a closed channel which is parabolic in x and uniform
358in y.  Test case 2 uses a gentler initial SSH slope which nevertheless demonstrates the
359ability to wet and dry on both sides of the channel. This solution requires values of
360$\mathrm{rn\_wdmin1}$ at least 0.3m ({\it Q.: A function of the maximum topographic
361slope?})
362
363\namdisplay{nam_wad_tc2}
364
365%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
366\begin{figure}[htb] \begin{center}
367\includegraphics[width=0.8\textwidth]{Fig_WAD_TC2}
368\caption{ \label{Fig_WAD_TC2}
369The evolution of the sea surface height in WAD\_TEST\_CASE 2 from the initial state (t=0)
370over the first three hours of simulation. Note that in this time-frame the resultant sloshing
371causes wetting and drying on both sides of the parabolic channel.}
372\end{center}\end{figure}
373%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
374
375\clearpage
376\subsubsection [WAD test case 3 : A parabolic channel (extreme slope) ]
377                    {WAD test case 3 : A parabolic channel (extreme slope)}
378\label{WAD_test_case3}
379
380Similar to test case 2 but with a steeper initial SSH slope. The solution is similar but more vigorous.
381
382\namdisplay{nam_wad_tc3}
383
384%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
385\begin{figure}[htb] \begin{center}
386\includegraphics[width=0.8\textwidth]{Fig_WAD_TC3}
387\caption{ \label{Fig_WAD_TC3}
388The evolution of the sea surface height in WAD\_TEST\_CASE 3 from the initial state (t=0)
389over the first three hours of simulation. Note that in this time-frame the resultant sloshing
390causes wetting and drying on both sides of the parabolic channel.}
391\end{center}\end{figure}
392%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
393
394\clearpage
395\subsubsection [WAD test case 4 : A parabolic bowl ]
396                    {WAD test case 4 : A parabolic bowl}
397\label{WAD_test_case4}
398
399Test case 4 includes variation in the y-direction in the form of a parabolic bowl. The
400initial condition is now a raised bulge centred over the bowl. Figure \ref{Fig_WAD_TC4}
401shows a cross-section of the SSH in the X-direction but features can be seen to propagate
402in all directions and interfere when return paths cross.
403
404\namdisplay{nam_wad_tc4}
405
406%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
407\begin{figure}[htb] \begin{center}
408\includegraphics[width=0.8\textwidth]{Fig_WAD_TC4}
409\caption{ \label{Fig_WAD_TC4}
410The evolution of the sea surface height in WAD\_TEST\_CASE 4 from the initial state (t=0)
411over the first three hours of simulation. Note that this test case is a parabolic bowl with
412variations occurring in the y-direction too (not shown here).}
413\end{center}\end{figure}
414%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
415
416\clearpage
417\subsubsection [WAD test case 5 : A double slope with shelf channel ]
418                    {WAD test case 5 : A double slope with shelf channel}
419\label{WAD_test_case5}
420
421Similar in nature to test case 1 but with a change in slope and a mid-depth shelf.
422
423\namdisplay{nam_wad_tc5}
424
425%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
426\begin{figure}[htb] \begin{center}
427\includegraphics[width=0.8\textwidth]{Fig_WAD_TC5}
428\caption{ \label{Fig_WAD_TC5}
429The evolution of the sea surface height in WAD\_TEST\_CASE 5 from the initial state (t=0)
430over the first three hours of simulation. The surge resulting in this case wets to the full
431depth permitted (2.5m above sea-level) and is only halted by the 4m high side walls.}
432\end{center}\end{figure}
433%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
434
435\clearpage
436\subsubsection [WAD test case 6 : A parabolic channel with central bar ]
437                    {WAD test case 6 : A parabolic channel with central bar}
438\label{WAD_test_case6}
439
440Test cases 1 to 5 have all used uniform T and S conditions. The dashed line in each plot
441shows the surface salinity along the y=17 line which remains satisfactorily constant. Test
442case 6 introduces variation in salinity by taking a parabolic channel divided by a central
443bar (gaussian) and using two different salinity values in each half of the channel. This
444step change in salinity is initially enforced by the central bar but the bar is
445subsequently over-topped after the initial SSH gradient is released. The time series in
446this case shows the SSH evolution with the water coloured according to local salinity
447values. Encroachment of the high salinity (red) waters into the low salinity (blue) basin
448can clearly be seen.
449
450\namdisplay{nam_wad_tc6}
451
452%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
453\begin{figure}[htb] \begin{center}
454\includegraphics[width=0.8\textwidth]{Fig_WAD_TC6}
455\caption{ \label{Fig_WAD_TC6}
456The evolution of the sea surface height in WAD\_TEST\_CASE 6 from the initial state (t=0)
457over the first three hours of simulation. Water is coloured according to local salinity
458values. Encroachment of the high salinity (red) waters into the low salinity (blue) basin
459can clearly be seen although the largest influx occurs early in the sequence between the
460frames shown.}
461\end{center}\end{figure}
462%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
463
464\clearpage
465\subsubsection [WAD test case 7 : A double slope with shelf, open-ended channel ]
466                    {WAD test case 7 : A double slope with shelf, open-ended channel}
467\label{WAD_test_case7}
468
469Similar in nature to test case 5 but with an open boundary forced with a sinusoidally
470varying ssh. This test case has been introduced to emulate a typical coastal application
471with a tidally forced open boundary. The bathymetry and setup is identical to test case 5
472except the right hand end of the channel is now open and has simple ssh and barotropic
473velocity boundary conditions applied at the open boundary. Several additional steps and
474namelist changes are required to run this test.
475
476\namdisplay{nam_wad_tc7}
477
478In addition, the boundary condition files must be generated using the python script
479provided.
480
481\begin{verbatim}
482python ./makebdy_tc7.py
483\end{verbatim}
484
485will create the following boundary files for this test (assuming a suitably configured
486python environment: python2.7 with netCDF4 and numpy):
487
488\begin{verbatim}
489  bdyssh_tc7_m12d30.nc   bdyuv_tc7_m12d30.nc
490  bdyssh_tc7_m01d01.nc   bdyuv_tc7_m01d01.nc
491  bdyssh_tc7_m01d02.nc   bdyuv_tc7_m01d02.nc
492  bdyssh_tc7_m01d03.nc   bdyuv_tc7_m01d03.nc
493\end{verbatim}
494
495These are sufficient for up to a three day simulation; the script is easily adapted if
496longer periods are required.
497
498%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
499\begin{sidewaysfigure}[htb] \begin{center}
500\includegraphics[width=0.8\textwidth]{Fig_WAD_TC7}
501\caption{ \label{Fig_WAD_TC7}
502The evolution of the sea surface height in WAD\_TEST\_CASE 7 from the initial state (t=0)
503over the first 24 hours of simulation. After the initial surge the solution settles into a
504simulated tidal cycle with an amplitude of 5m. This is enough to repeatedly wet and dry
505both shelves.}
506
507\end{center}\end{sidewaysfigure}
508%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
509
510
511% ================================================================
512
513\bibliographystyle{wileyqj}
514\bibliography{references}
515
516\end{document}
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