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On the effect of bathymetry in numerical storm surge simulation experiments
Computersand Fluids Vol. 11, No. 3, pp, 161-174, 1983 Printed in Great Britain.
0045-7930/83/030161-14503.00/0 Pergamon Press Ltd.
ON THE EFFECT OF BATHYMETRY IN NUMERICAL STORM SURGE SIMULATION EXPERIMENTS B. JOHNS Departmentof Meteorology,Universityof Reading,Reading,England and P. C. SINHA,S. K. DUBE, U. C. MOHANTYand A. D. RA0 Centre for AtmosphericSciences,IndianInstituteof Technology,Delhi,India (Received 15 March 1982;in revised form 18 August 1982)
~ e t - - A numericalmodel is describedfor the simulationof storm surges which uses a non-uniform off-shoregrid-spacingadjacentto coastal boundaries.This permitsan increasedresolutionnearthe coast in the modelsdescribedin[l, 2]. Usingdata on the 1977Andhracyclone,whichstruck the east coast of India, it is shownthat the near-coastalbathymetryis criticalin determiningthe coastal surge-inducedsea-surface elevation.It is also shownthat the coastalsurgeresponseis effectivelyindependentof the depthof waterin the deepestregionsof the analysis area. Conclusionsare drawn concerningthe selectionof an optimum resolutionin the numericalschemetogetherwith the properrepresentationof the bathymetry. INTRODUCTION In two recent papers, Johns et al. [1, 2] have described numerical models for the simulation of storm surges along the east coast of India. In particular, these have been applied to the surge generated by the cyclone which struck the coast of Andhra Pradesh in November, 1977. From the numerical point of view, an important feature of these models is the use of a staggered finite-difference grid on which the sea-surface elevation and components of velocity are carried at different computational points. With this scheme, sea, surface elevations are not directly computed at points along the coastline. In order to derive estimates of the coastal surge elevation, it is necessary to extrapolate from computed elevations at those adjacent off-shore grid-points at which they are carried. A consequence of this procedure is that the equilibrium depth of water at the coastline (which, in the case of [1], is represented by a fixed vertical side-wall) is not actually used in the computation. The minimum depths used in the numerical scheme will in fact be those at the first off-shore grid-points. In the simulations reported in [1,2], the grid-spacing across the coastal zone is uniform and relatively coarse. However, the sea-floor off the east coast of India consists of a steeply sloping shelf over which the depth of water increases rapidly in the seaward direction. With a grid-spacing that is too coarse, this means that the real bathymetry may lead to a depth of water at the first off-shore points that is considerably in excess of that at points nearer the coastline. We have already noted that the extrapolated coastal surge elevation is dependent on the elevation computed directly at these off-shore points. Therefore, since the surge elevation is known to increase with a decrease in the equilibrium depth of water, we have to recognize that an injudicious use of our scheme may lead to an underestimate of the elevation at coastal stations. The procedure adopted in [1] to compensate for a possibly inadequate resolution of the near-coastal bathymetry involved the use of notional reduced depths at the first off-shore grid-points. These are intended to be representative of the average of the actual depth in the zone immediately adjacent to the coastline. In fact, our simulations compared reasonably well with available observational data but the procedure should be further supported by an approach depending on a more detailed specification of the actual near-coastal bathymetry. An overall increase in the resolution leads to a substantial increase in the computational overheads and is, in any case, unnecessary far away from the coastline. Accordingly, in the present paper, we have extended the methods described in [1] to allow for a grid-refinement in the near-coastal zone. In contrast with [3-5], for example, our method does not depend on the patching together CAF Vol, ll, No. 3--A
161
B. JOHNSet aL
162
of computational regions having different uniform grid-spacings. It is more akin to work reported by Sundqvist and Veronis[6] wherein a continuously contracting grid-increment is used to resolve the coastal boundary layer in a wind-driven ocean circulation model. We have used the grid-refinement model in experiments designed to determine the effect on the coastal surge response of variations in the representation of the near-coastal bathymetry. The results of these experiments support our earlier procedure in which we used a relatively coarse and uniform coastal zone grid-spacing. We have also applied the model to determine the sensitivity of the computed coastal surge elevation to changes in the specification of the bathymetry in the deep-water regions of the analysis area. This experiment shows that the coastal surge elevation is effectively independent of the actual depth of water in the deep-water regions for depths in excess of about 500 m. The use of actual depths -300 m has no effect on the computed surge response and serves only to increase the computational overheads as a result of the reduction in the time-step that is required to maintain computational stability. 2. F O R M U L A T I O N
As in [1], we use a set of rectangular Cartesian axes in which the origin, O, is within the equilibrium level of the sea-surface. Ox points towards the east, Oy towards the north and Oz is directed vertically upwards. The displaced position of the sea-surface is given by z = ~(x, y, t) and the position of the sea-floor by z = - h(x, y). A western coastal boundary (the east coast of India) is situated at x = bl(y) and an eastern open-sea boundary is at x = b2(y). Southern and northern open-sea boundaries are at y = 0 and y = L, respectively. This configuration is shown in Fig. 1. The depth-averaged components of velocity, u and v, then satisfy c3u ~gu a~" 1 ~ /)2)1/2} ~xx + V - ~ - [ v = - g ~ x + ~ p {Tx -k#u(u2+
-cgu -+v
at
(2.1)
and
-
Ov + Ot -
8v Ov + c9~" . 1 , ~ u -~x + v - ~ [u = - g - ~ t Hpp i T ' -
kpv(u 2 + v2),/2}.
(2.2)
25"
y--L "':.." BANGLADESH
INDIA
/ ~ T...'7 ' / / / ~
:.
",%.
///"
:x.=b,(y) , ~ /
20"
" BURMA . :_ i :" '-
,',,
:.......:..."
'.. M
"., [
15"
" -'" 10"
rf / L
',',
~..-..
:'......'"L I o
. _x _ I
80"
85"
_
_~ x
I
90" Fig. 1. The analysis area.
I
95"
5"
On the effectof bathymetryin numericalstormsurgesimulationexperiments
163
In (2.1) and (2.2), f denotes the Coriolis parameter, the pressure is taken as hydrostatic and H is the total depth ~ + h. (~-xc, ~-y:)denotes the applied surface wind-stress and the bottom stress is parameterized in terms of a quadratic law. p denotes the water density and the friction coefficient, k, is taken as 2.6 × 10-3. In the present study, the position of the lateral boundaries is invariant in time. Following [1, 2], we therefore obtain the lateral boundary conditions u - v -abl ~y = 0
u-
at x = bl(y)
a t x = b2(y)
v Ob2-(~)u2~=O 0y
v+
(2.3)
(2.4)
~'=0 a t y = O
(2.5)
I/2
The vertically integrated form of the equation of continuity is OH O -~- + - ~ (Hu)+
~y (Hv)=
0.
(2.7)
The coordinate transformation described in [1] is based upon a new set of independent variables, ~ y and t where bl(y) ~= x -b(y)
(2.8)
b(y) = b2(y) - bz(y).
(2.9)
It then follows that (2.1) and (2.2) may be transformed into
a~ a , a__ a~ + b~x~ k~, 2+v2)U2 - ~ + -~ ( UEO+ ay ( vEO - fE = - g H -~ P "~ t u
(2.10)
and
o~+ ~a, ..... ] be.¢ k~. 2+ v2),/2, -~ ~ + ~a( ~ ) + / ~ = - g H (b ~a¢- ~ (ab,+~sab]a¢ ~/~j+p,--g~. (2.11) where (0bl bU= u - \ - ~ +
ab) ~Ty v
= Hbu = Hbv J"
(2.12)
(2.13)
The boundary conditions (2.3) and (2.4) become U=0
b(y)U -
ate=0
(2.14)
~ = 0 at ~:= 1.
(2.15)
164
B. JOHNSet al.
Equation (2.7) is readily found to yield (2.16)
A discretization procedure was described in [1] in which a uniform spacing, AE, was taken between grid-points in the direction of the E-coordinate. This led to a scheme of numerical solution in which the coastal boundary could be modelled with considerable realism. However, a desirable feature of storm surge simulation schemes is the ability to incorporate increased resolution adjacent to the coastline. This facility is not present in [1] but it may be introduced by making a further transformation of the E-coordinate. This is achieved by defining a new variable, 7, by
7 = E+ ~ln (E+ E°), ~0
(2.17)
where e and E0 are disposable parameters. By virtue of (2.17), we see that finite increments AE and AT/are related by A7 AE = 1 + d(E + Eo)"
(2.18)
If we consider a uniform value of A7 and, for example, take ~ = 0.04andEo= 0.001, we find, as A T e 0 , that (AOe~0 = (0.024)A7 (AO~=0.5= (0.926)An (A0¢=1 = (0.962)An.
(2.19)
Thus, near the coastline, the transformation (2.17) will lead to a substantial mesh refinement in which the grid-increment is reduced to a fraction of its essentially uniform value between E = 0.5 and E = 1.0. Therefore, the use of (2.17) in (2.10), (2.11) and (2.16) will assist in the incorporation of a more detailed bathymetric specification in the important coastal region. It must, however, be emphasized that the use of such a transformation is not without attendant danger. Equation (2.17) would be a natural transformation to use if the solution were known to have an almost linear dependence on 71. This, of course, is not necessarily the case and an injudicious choice of values for • and E0 could lead to results contaminated by a substantial truncation error. It may also be noted that the use of (2.17) will accommodate a surge response having a rapid off-shore spatial variation characteristic of a coastally-trapped wave. An exponentially trapped wave would have the form E = A(y, t) e -e",
(2.20)
where l is the off-shore e-folding scale. A numerical treatment of such a response must necessarily resolve this scale and this requires that (af)~o ~ 1.
(2.21)
a__~= F(7) = l an 1 + ~/(f + to)'
(2.22)
1
Writing
On the effectof bathymetryin numericalstormsurgesimulationexperiments
165
equations (2.10), (2.11) and (2.16) take the form
OFt+ 1 0 . . . .
_~y
gH a~ ÷ M', ~. k~ (u 2 +
--
~-\0y
v2)112'
(2.23)
~c~ylF0rlj+ P --~(u + (2.24)
and
-~ (b~)+ ff1 ~0 (bHU) + -~y(bHv)
=
O.
(2.25)
3. NUMERICAL SOLUTION The discretization procedure and the finite-difference grid selected for the numerical solution of (2.23)-(2.25) are completely analogous to those reported in detail in[l, 2]. It is therefore unnecessary to repeat here the full account given earlier. We write 7 =~=(i-1)A~,
i=1,2 .... m
(3.1)
and define AT by Art = rl,n](m - 1),
(3.2)
where ~Tmis determined from (2.17) with ~ = 1, thus yielding 7,, = 1 + ~ In (1 + 1/~:o).
(3.3)
We also write Y = Yi = (1- DAY, J = 1, 2. . . . n;
Ay = L/(n - 1)
(3.4)
and define a sequence of time-instants by
t=tp=pAt,
p=0,1 ....
(3.5)
Discrete forms of (2.23)-(2.25) are then written on the type of staggered grid used in [1, 2]. The only difference relates to the factor of F appearing in the basic equations. This, of course, must be evaluated at each of the discrete ,/-points given by (3.1). To do this, we use (2.17) to determine the value of ~ at each of the discrete ~/-points by applying a NewtonRaphson iterative procedure. The corresponding discrete values of F are readily deduced from (2.22) and these may then be substituted into the basic finite-difference equations to be used in the updating of ~, u and v. 4. NUMERICALEXPERIMENTATION Numerical experiments are performed using the analysis area shown in Fig. 1. On average, the width of the coastal zone is about 300km and an idealized cyclone moves along the indicated north-westerly track for about 3 days before landfall at the coast of Andhra Pradesh. Thus, all conditions are identical to those prescribed in [1,2]. The idealized cyclonic wind field is simulated as in [2] by representing the azimuthal wind speed in the form Vo(r/R) /2
V=
for r ~
Voexp(R.~_£)forr>R,
(4.1)
166
B. JOHNSet al.
where Vo = 70 m s-1, R = 80 km and c = 240 km. The associated wind stress is calculated from a quadratic law with a friction coefficient of 2.8 x 10 -3. In this paper, we have taken a basic setting in which m = 10, n = 49, • = 0.04, ~0= 0.001 (resolution R1). Thus, ~,~ = 1.28 and At/-0.14 and corresponding discrete values of ~ and ,/are given in Table 1. From this, we note that the first off-shore grid-point at which the elevation is computed is, on average, about 6 km from the coastline. The estimated value of the elevation at a coastal station is derived by linear extrapolation from the elevation at the first off-shore point and that situated, on average, about 60 km from the coastline. In our earlier numerical simulations employing a uniform grid-spacing across the coastal zone the linear extrapolation to coastal stations was based upon elevations computed at grid-points respectively about 30 and 90 km from the coastline. The present technique therefore yields a much refined numerical scheme in which account may be taken of more detailed variations in the near-coastal bathymetry. The increased coastal resolution is obtained at the cost of a coarser resolution in the deep water adjacent to the eastern open-sea boundary. In this region, as shown later, variations in the bathymetry are of less importance than in shallow water although the resolution must still be adequate to resolve the scale of the forcing cyclonic wind pattern. Our object in the experimentation is to investigate the dependence of the surge response on variations in the bathymetric specification. For this purpose, it is convenient and, with an appropriate choice of parameters, reasonably realistic to represent the gross bathymetry in the Bay of Bengal by a function of the form y (y/L)'/
(4.2) In (4.2), a,/3 and 3' are disposable constants and ho is a depth scale that corresponds to the equilibrium depth of water along the coastline ~ = 0 and at y = L. Thus, values of a a n d / / m a y be selected to model the maximum depth of water at the south-eastern extremity of the analysis area (~ = l, y = 0) together with the off-shore slope of the sea-floor. The value of 3' may be chosen to model the northward rate of reduction in the depth of the water. Equation (4.2) is then applied to generate the bathymetry at the discrete q-points by using numerically inverted values of ~ obtained from (2.17).
Table 1. Discrete values of ~ and ~ with resolution RI. A~?= 0.14. nl~n
o
o
1
0.020
2
0.099
3
0.211
4
0.335
5
0.463
6
0.595
7
0. 729
8
0.864
9
! .000
On the effectof bathymetryin numericalstormsurgesimulationexperiments
167
(i) Comparison of non-uniform and uniform grid models In this experiment, we have used the grid-refinement model (M1) with a bathymetry B1 corresponding to a -- 4.6052,/3 = 0.25, 3' -- 0.5, ho = 5 m in (4.2) together with the resolution R1. This yields a fairly realistic approximation to the gross bathymetry in the immediate off-shore region along the east coast of India but it considerably underestimates the maximum depths actually attained-particularly those in the south-eastern sector of the analysis area. With B1, the maximum depth of water is 500 m at ~:= 1, y = 0. At ~ = 0.02, y = .)27(about 7.6 km off-shore of Kavali), h = 13.26 m, whilst at ~ = 1/9, y = Y27,the depth is about 69% greater and equal to 22.35 m. We compare results from M1 with those from a model M2 (obtained by putting e = 0 in (2.17)) which corresponds to the type of coastal zone model described in [1] in which the scaled off-shore grid increment is uniform across the coastal zone and equal to 1/9. Two different bathymetries are used in experiments with M2. The first is B 1 whils*,the second is B2 and corresponds to a = 3.92, /3 = 0.75, y = 1.0, ho = 10 m. B2 reproduces almost exactly the bathymetry used in [1] and shown in Fig. 3 of that paper. It implies a maximum depth of 500 m at ~:= 1, y = 0 but at ~:= 1/9, y = Y27, h = 16.39 m. This is about 27% less than that at the same position with B1 but is within 5% of the average depth of approximately 15.6 m obtained with B1 for 0<~:< 1/9. The model bathymetric sections off Kavali calculated from B 1 and B2 are shown in Fig. 2. In Fig. 3, we give the time-history of the sea-surface elevation at Kavali and Divi as determined from M1 with R1 and M2 when each of these has bathymetry B1. In comparison with M2, we note that M1 produces a higher peak surge at both stations. At Kavali, the peak surge response calculated from M1 is about 29% higher than that computed from M2. This indicates that the depth of water at the first off-shore grid-points is critical in determining the surge at coastal stations. In this connection, the adequacy of the near-coastal bathymetric resolution in M1 has been investigated by running MI(B1) with m = 12, n =49, ~0=0.001, = 0.04 (resolution R2). At y = Y27,we note that the first off-shore grid-point is approximately 4.7 km from the coastline and the corresponding equilibrium depth is about 11.87 m. With R2, the depth of water at the first off-shore point is therefore reduced by about 45% compared with R1. However, at t = 55 hr, we find that the surge response at Kavali is within 2% of that obtained with R1. We infer from this result that R1 leads to a satisfactory resolution of the near-coastal bathymetry; by moving the first off-shore grid-point nearer the coastline, there is no significant change in the computed surge response at Kavali.
0
5O I.UJ
\ \
Z
\
"r
1oo
\
uJ
\ \
,50
0
I
0.25
I
0.5
I
0.75
~
1.0
Fig. 2. Model bathymetric sections off Kavali. Continuous line is from B 1. Broken line is from B2.
168
B. JOHNSet al.
5.0
OlVl
KAVALI
40 30 2.0 L~J CE
2"s- ~.0 Z
z
0
O
"~ -~.o uJ w
-2.0 -3.0 -40 -5.0 -6.0 -7.0 -8.0
I
l
I
I
I
0
20
40
60
80
100
TIME IN HOURS
Fig. 3. Surge elevations at Kavali and Divi computed from MI(BI) (continuous line) with resolution RI and from M2(B1) (broken line).
We next compared responses obtained from MI(B1) using R1 with those from M2(B2). The time-histories of the sea-surface elevation at Kavali and Divi are given in Fig. 4. We note that both models produce almost the same peak response at Kavali. With the coarser (and uniform) coastal zone resolution in M2 this supports our view that it is necessary to adjust appropriately the bathymetry at the first off-shore grid-points. This adjustment takes the form of the use of notional reduced depths that are more in accord with the average depth for 0<~ ~< 1/9 in MI(B1). Whilst this procedure is not necessarily recommended as an alternative to using MI(BI), it presumably explains why the simulations given in [1] were in reasonable agreement with available data derived from reports of flooding along part of the coast of Andhra Pradesh. (ii) Convergence studies In order to estimate the effect of overall truncation error in the numerical scheme, we next ran MI(B1) with m = 14, n = 49, to = 0.001, ~ = 0.0226 (resolution R3) and give the corresponding discrete values of ~ and ~ in Table 2. With R3, the positions of the first off-shore grid-points coincide exactly with those resulting from the use of R1. Accordingly, a precise comparison may be made between directly computed elevations at ~ = 0.02 with two different coastal zone resolutions. For ~> 0.02, we note from Table 2 that R3 leads to a coastal zone resolution increased by between 65 and 80% compared with RI. We are therefore able to check that the forcing wind-stress field is being properly resolved in the model. In Fig. 5, we give the distribution of the maximum sea-surface elevation (peak surge envelope) together with its time of occurrence at ~ = 0.02, 0 < y < L. These are computed from MI(B1) with both R1 and R3. For (y/L)> 0.4, it is noteworthy that the peak surge propagates parallel to the coastline with a phase that generally becomes later with increasing y.
ELEVATION
,'~ 'TJ
~.~"
IN M E T R E S
o
I
I
I xl
I ¢D
E
o
ELEVATION IN METRES
L,'n
~
o~
r
E~"
x
~
o
o
0
o
0
0
I
I
I
I
I
I
I
o
~
o
o
o
o
o
I
I
I
J
[
~-
E
=t ]~
~g
8~
o
r-
o E. "I"1
~°~
('I)
~.~ II o
0
A ~I",
TIME
IN
HOURS
c,~
B. JOHNS et al.
170
Table 2. Discrete values of ~ and "Owith resolution R3, A'0 0.089. =
n/An o
o
1
0.020
2
0.079
3
0.153
4
0.232
5
0.315
6
O. 398
7
0.483
8
O. 568
9
0.654
I0
0.740
11
0.826
12
0.913
13
1.000 i
In Fig. 6, the instantaneous level of the sea-surface is shown off Kavali at t = 50 h and t = 100h as a function of the scaled off-shore distance ~. These two instants correspond respectively to positive and negative phases of the coastal surge and are calculated using a resolution R5 given by m = 30, n = 49, ~0= 0.001, e = 0.0015. We see that the surge response has the characteristics of a northward propagating coastatly-trapped wave as described, e.g. in [7]. For the positive phase, the off-shore e-folding scale is given by l = 0.35 and so (A~)~=0//= 0.057 and (2.21) is clearly satisfied. The extrapolated surge envelope at coastal stations shows that peak elevations exceeding 1 m are predicted over a north-south distance of almost 475 km. This envelope may be atypical for surges along the east coast of India and is comparable with that reported for the surge generated by Hurricane Carla in 1961 in the Gulf of Mexico. In this, surges exceeding 1 m were monitored from Texas to the coast of Louisiana--a distance of over 800 km [8]. Further to our suggestion that the surge response consists of a northward travelling coastally-trapped wave, we refer to work reported by Heaps [9] on the numerical simulation of externally generated surges in the North Sea. The inference here is that the surge response propagates southwards along the east coast of the British Isles in the form of a trapped Kelvin wave. In this, the confinement of the peak surge energy to the coastal regions is a consequence of the earth's rotation. In contrast with this, a northward propagation of Kelvin waves along the east coast of India is not possible. Hence, a northward propagating coastally-trapped wave in this region must necessarily be related to the existence of the steep off-shore slope of the sea-floor. The nature of the response therefore appears related to the northward propagation of a species of edge wave as studied by Reid[10]. We note from Fig. 5 that the increased resolution obtained with R3 does not significantly alter the peak surge elevation at ~ = 0.02 with regard to either its amplitude or phase. The root
On the effect of bathymetry in numerical storm surge simulation experiments
171
0 2O
z 60 80 ° z 0 --
.~ .i
/" -2 - -
/ /
/ -4 0
I
025
I
0.5
I
0 35
f
1.0
Fig. 6. Instantaneous elevation, ~, of sea-surface off Kavali. ~ 6 at t = 5 0 h ; . . . . . . . t = 100 K . . . . corresponds to bathymetric section off Kavali computed from BI.
I0~ at
mean square difference between computed elevations at ~ = 0.02 (0 ~
172
B. JOHNS et al.
5.0 I l I
4.0
3.0
I
\,
-
I i I I
DIVI
,, ;..//
2.0 uJ
~1.0 W
_
Z
z
\,,
,, ,,/,/q ./
t / /
,
\
0
t~ -1.0 .._1 UA
!
-2.0
_
! !
-30
~4.0
-5.0
-6.0
-7.0
I
20
I
z.,0
1
~"
60 TIME IN HOURS
80
I
~00
Fig. 7. Surge elevations at Kavali and Divi computed from M2(B2) (broken line) and M2(B3) (continuous
line).
5. C O N C L U S I O N S
We have investigated the effect of different finite-difference resolutions, and their relation to the bathymetric specification, in a numerical storm surge model applicable to the east coast of India. It is shown that the proper resolution of the immediate near-coastal bathymetry is critical in determining the storm-induced sea-surface elevation. We have described a method by means of which the necessary resolution can be obtained. This involves a near-coastal continuous grid compression. Experiments using this grid refinement model have also been performed to investigate the necessary resolution for the representation of the spatial scale of the forcing cyclone. The track followed by the forcing cyclone in this investigation has a substantial northward long-shore component prior to the landfall of the storm. As a result of the experiments, we suggest that the associated surge response is predominantly a result of the northward propagation of a coastally-trapped wave together with a contribution from local wind-stress forcing. The near-coastal local amplitude of the trapped wave is strongly dependent on the local bathymetry and coastline c o ~ a t i o n . The geographical position of the peak surge elevation is therefore determined by a complicated dynamical process. This appears to contrast with the case of the almost normal incidence of a storm track on a straight coastline where, we suggest,
On the effect of bathymetry in numerical storm surge simulation experiments
5.0
173
--
KAVALI
L,.0
I
3.0
-
2.0
-
1.0
U3 ttl t',,' uJ
0
z z o
- I .0
-2.0 -.I UJ
-3.0
-40 -50 -6.0 ~7.0
20
/.0 TIME
60 IN HOURS
80
100
Fig. 8. Surge elevations at Kavali and Divi computed from MI(BI) (continuous line) and MI(B4) (broken line).
the surge response is predominantly a consequence of local wind-stress forcing. In this case, there is hardly any contribution from an alongshore propagating coastally-trapped wave. We have also shown that the coastal surge response is independent of the bathymetry in the far off-shore deep water regions. This is partly explained by the fact that the off-shore distance of these regions greatly exceeds the off-shore e-folding scale of the coastally-trapped wave. Acknowledgement--A large part of this work was carried out with support from the U.K. Overseas Development Administration originally under the auspices of the Imperial College Delhi Committee. REFERENCES 1. B. Johns, S. K. Dube, U. C. Mohanty and P. C. Sinha, Numerical simulationof the surge generated by the 1977Andhra cyclone. Q. Z It. Met. Soc. 107, 919-934 (1981). 2. B. Johns, S. K. Dube, P. C. Sinha, U. C. Mohanty and A. D. Rao, The simulation of a continuously deforming laterial boundary in problems involving the shallow water equations. Computers and Fluids 10(2), 105-116(1982). 3. C. P. Jelesnianski, A numerical calculation of storm tides induced by a tropical storm impingingon a continental shelf. Moil Weath. Rev. 93, 343-358 (1965). 4. P. K. Das, M. C. Sinha and V. Balasubramanyam,Storm sngres in the Bay of Bengal. (2..£ It. Met. Soc. leO, 437-449 (1974). 5. B. Johns and M. A. Ali, The numerical modellingof storm surges in the Bay of Bengal: (2. J, R. Met. Soc. 106,1-18 (1980).
174
B. JOHNSet aL
6. H. Sundqvist and G. Veronis, A simple finite-diffeiencegrid with non-constant intervals. Tellus 22, 26-31 (1970). 7. P. H. Leblond and L. A. Mysak, Trapped coastal waves and their role in shelf dynamics. In "The Sea" (Edited by E. Golberg, I. McCave, J. O'Brien and J. Steele). Vol. 6, pp. 459--495Wiley-lnterscience,New York (1977). 8. World MeteorologicalOrganizationreport on marine science affairs No. 13. Present techniques of tropical storm surge prediciton, p. 87 (1978). 9. N. S. Heaps A two-dimensionalnumerical sea model. Phil. Trans. R. $oc., Lond. A265, 93--137.(1969). 10. R. O, Reid, Effect of Coriolis force on edge-waves. (1) Investigation of normal modes. J. Mar. Res. 16, 109-141. (1958).
0045-7930/83/030161-14503.00/0 Pergamon Press Ltd.
ON THE EFFECT OF BATHYMETRY IN NUMERICAL STORM SURGE SIMULATION EXPERIMENTS B. JOHNS Departmentof Meteorology,Universityof Reading,Reading,England and P. C. SINHA,S. K. DUBE, U. C. MOHANTYand A. D. RA0 Centre for AtmosphericSciences,IndianInstituteof Technology,Delhi,India (Received 15 March 1982;in revised form 18 August 1982)
~ e t - - A numericalmodel is describedfor the simulationof storm surges which uses a non-uniform off-shoregrid-spacingadjacentto coastal boundaries.This permitsan increasedresolutionnearthe coast in the modelsdescribedin[l, 2]. Usingdata on the 1977Andhracyclone,whichstruck the east coast of India, it is shownthat the near-coastalbathymetryis criticalin determiningthe coastal surge-inducedsea-surface elevation.It is also shownthat the coastalsurgeresponseis effectivelyindependentof the depthof waterin the deepestregionsof the analysis area. Conclusionsare drawn concerningthe selectionof an optimum resolutionin the numericalschemetogetherwith the properrepresentationof the bathymetry. INTRODUCTION In two recent papers, Johns et al. [1, 2] have described numerical models for the simulation of storm surges along the east coast of India. In particular, these have been applied to the surge generated by the cyclone which struck the coast of Andhra Pradesh in November, 1977. From the numerical point of view, an important feature of these models is the use of a staggered finite-difference grid on which the sea-surface elevation and components of velocity are carried at different computational points. With this scheme, sea, surface elevations are not directly computed at points along the coastline. In order to derive estimates of the coastal surge elevation, it is necessary to extrapolate from computed elevations at those adjacent off-shore grid-points at which they are carried. A consequence of this procedure is that the equilibrium depth of water at the coastline (which, in the case of [1], is represented by a fixed vertical side-wall) is not actually used in the computation. The minimum depths used in the numerical scheme will in fact be those at the first off-shore grid-points. In the simulations reported in [1,2], the grid-spacing across the coastal zone is uniform and relatively coarse. However, the sea-floor off the east coast of India consists of a steeply sloping shelf over which the depth of water increases rapidly in the seaward direction. With a grid-spacing that is too coarse, this means that the real bathymetry may lead to a depth of water at the first off-shore points that is considerably in excess of that at points nearer the coastline. We have already noted that the extrapolated coastal surge elevation is dependent on the elevation computed directly at these off-shore points. Therefore, since the surge elevation is known to increase with a decrease in the equilibrium depth of water, we have to recognize that an injudicious use of our scheme may lead to an underestimate of the elevation at coastal stations. The procedure adopted in [1] to compensate for a possibly inadequate resolution of the near-coastal bathymetry involved the use of notional reduced depths at the first off-shore grid-points. These are intended to be representative of the average of the actual depth in the zone immediately adjacent to the coastline. In fact, our simulations compared reasonably well with available observational data but the procedure should be further supported by an approach depending on a more detailed specification of the actual near-coastal bathymetry. An overall increase in the resolution leads to a substantial increase in the computational overheads and is, in any case, unnecessary far away from the coastline. Accordingly, in the present paper, we have extended the methods described in [1] to allow for a grid-refinement in the near-coastal zone. In contrast with [3-5], for example, our method does not depend on the patching together CAF Vol, ll, No. 3--A
161
B. JOHNSet aL
162
of computational regions having different uniform grid-spacings. It is more akin to work reported by Sundqvist and Veronis[6] wherein a continuously contracting grid-increment is used to resolve the coastal boundary layer in a wind-driven ocean circulation model. We have used the grid-refinement model in experiments designed to determine the effect on the coastal surge response of variations in the representation of the near-coastal bathymetry. The results of these experiments support our earlier procedure in which we used a relatively coarse and uniform coastal zone grid-spacing. We have also applied the model to determine the sensitivity of the computed coastal surge elevation to changes in the specification of the bathymetry in the deep-water regions of the analysis area. This experiment shows that the coastal surge elevation is effectively independent of the actual depth of water in the deep-water regions for depths in excess of about 500 m. The use of actual depths -300 m has no effect on the computed surge response and serves only to increase the computational overheads as a result of the reduction in the time-step that is required to maintain computational stability. 2. F O R M U L A T I O N
As in [1], we use a set of rectangular Cartesian axes in which the origin, O, is within the equilibrium level of the sea-surface. Ox points towards the east, Oy towards the north and Oz is directed vertically upwards. The displaced position of the sea-surface is given by z = ~(x, y, t) and the position of the sea-floor by z = - h(x, y). A western coastal boundary (the east coast of India) is situated at x = bl(y) and an eastern open-sea boundary is at x = b2(y). Southern and northern open-sea boundaries are at y = 0 and y = L, respectively. This configuration is shown in Fig. 1. The depth-averaged components of velocity, u and v, then satisfy c3u ~gu a~" 1 ~ /)2)1/2} ~xx + V - ~ - [ v = - g ~ x + ~ p {Tx -k#u(u2+
-cgu -+v
at
(2.1)
and
-
Ov + Ot -
8v Ov + c9~" . 1 , ~ u -~x + v - ~ [u = - g - ~ t Hpp i T ' -
kpv(u 2 + v2),/2}.
(2.2)
25"
y--L "':.." BANGLADESH
INDIA
/ ~ T...'7 ' / / / ~
:.
",%.
///"
:x.=b,(y) , ~ /
20"
" BURMA . :_ i :" '-
,',,
:.......:..."
'.. M
"., [
15"
" -'" 10"
rf / L
',',
~..-..
:'......'"L I o
. _x _ I
80"
85"
_
_~ x
I
90" Fig. 1. The analysis area.
I
95"
5"
On the effectof bathymetryin numericalstormsurgesimulationexperiments
163
In (2.1) and (2.2), f denotes the Coriolis parameter, the pressure is taken as hydrostatic and H is the total depth ~ + h. (~-xc, ~-y:)denotes the applied surface wind-stress and the bottom stress is parameterized in terms of a quadratic law. p denotes the water density and the friction coefficient, k, is taken as 2.6 × 10-3. In the present study, the position of the lateral boundaries is invariant in time. Following [1, 2], we therefore obtain the lateral boundary conditions u - v -abl ~y = 0
u-
at x = bl(y)
a t x = b2(y)
v Ob2-(~)u2~=O 0y
v+
(2.3)
(2.4)
~'=0 a t y = O
(2.5)
I/2
The vertically integrated form of the equation of continuity is OH O -~- + - ~ (Hu)+
~y (Hv)=
0.
(2.7)
The coordinate transformation described in [1] is based upon a new set of independent variables, ~ y and t where bl(y) ~= x -b(y)
(2.8)
b(y) = b2(y) - bz(y).
(2.9)
It then follows that (2.1) and (2.2) may be transformed into
a~ a , a__ a~ + b~x~ k~, 2+v2)U2 - ~ + -~ ( UEO+ ay ( vEO - fE = - g H -~ P "~ t u
(2.10)
and
o~+ ~a, ..... ] be.¢ k~. 2+ v2),/2, -~ ~ + ~a( ~ ) + / ~ = - g H (b ~a¢- ~ (ab,+~sab]a¢ ~/~j+p,--g~. (2.11) where (0bl bU= u - \ - ~ +
ab) ~Ty v
= Hbu = Hbv J"
(2.12)
(2.13)
The boundary conditions (2.3) and (2.4) become U=0
b(y)U -
ate=0
(2.14)
~ = 0 at ~:= 1.
(2.15)
164
B. JOHNSet al.
Equation (2.7) is readily found to yield (2.16)
A discretization procedure was described in [1] in which a uniform spacing, AE, was taken between grid-points in the direction of the E-coordinate. This led to a scheme of numerical solution in which the coastal boundary could be modelled with considerable realism. However, a desirable feature of storm surge simulation schemes is the ability to incorporate increased resolution adjacent to the coastline. This facility is not present in [1] but it may be introduced by making a further transformation of the E-coordinate. This is achieved by defining a new variable, 7, by
7 = E+ ~ln (E+ E°), ~0
(2.17)
where e and E0 are disposable parameters. By virtue of (2.17), we see that finite increments AE and AT/are related by A7 AE = 1 + d(E + Eo)"
(2.18)
If we consider a uniform value of A7 and, for example, take ~ = 0.04andEo= 0.001, we find, as A T e 0 , that (AOe~0 = (0.024)A7 (AO~=0.5= (0.926)An (A0¢=1 = (0.962)An.
(2.19)
Thus, near the coastline, the transformation (2.17) will lead to a substantial mesh refinement in which the grid-increment is reduced to a fraction of its essentially uniform value between E = 0.5 and E = 1.0. Therefore, the use of (2.17) in (2.10), (2.11) and (2.16) will assist in the incorporation of a more detailed bathymetric specification in the important coastal region. It must, however, be emphasized that the use of such a transformation is not without attendant danger. Equation (2.17) would be a natural transformation to use if the solution were known to have an almost linear dependence on 71. This, of course, is not necessarily the case and an injudicious choice of values for • and E0 could lead to results contaminated by a substantial truncation error. It may also be noted that the use of (2.17) will accommodate a surge response having a rapid off-shore spatial variation characteristic of a coastally-trapped wave. An exponentially trapped wave would have the form E = A(y, t) e -e",
(2.20)
where l is the off-shore e-folding scale. A numerical treatment of such a response must necessarily resolve this scale and this requires that (af)~o ~ 1.
(2.21)
a__~= F(7) = l an 1 + ~/(f + to)'
(2.22)
1
Writing
On the effectof bathymetryin numericalstormsurgesimulationexperiments
165
equations (2.10), (2.11) and (2.16) take the form
OFt+ 1 0 . . . .
_~y
gH a~ ÷ M', ~. k~ (u 2 +
--
~-\0y
v2)112'
(2.23)
~c~ylF0rlj+ P --~(u + (2.24)
and
-~ (b~)+ ff1 ~0 (bHU) + -~y(bHv)
=
O.
(2.25)
3. NUMERICAL SOLUTION The discretization procedure and the finite-difference grid selected for the numerical solution of (2.23)-(2.25) are completely analogous to those reported in detail in[l, 2]. It is therefore unnecessary to repeat here the full account given earlier. We write 7 =~=(i-1)A~,
i=1,2 .... m
(3.1)
and define AT by Art = rl,n](m - 1),
(3.2)
where ~Tmis determined from (2.17) with ~ = 1, thus yielding 7,, = 1 + ~ In (1 + 1/~:o).
(3.3)
We also write Y = Yi = (1- DAY, J = 1, 2. . . . n;
Ay = L/(n - 1)
(3.4)
and define a sequence of time-instants by
t=tp=pAt,
p=0,1 ....
(3.5)
Discrete forms of (2.23)-(2.25) are then written on the type of staggered grid used in [1, 2]. The only difference relates to the factor of F appearing in the basic equations. This, of course, must be evaluated at each of the discrete ,/-points given by (3.1). To do this, we use (2.17) to determine the value of ~ at each of the discrete ~/-points by applying a NewtonRaphson iterative procedure. The corresponding discrete values of F are readily deduced from (2.22) and these may then be substituted into the basic finite-difference equations to be used in the updating of ~, u and v. 4. NUMERICALEXPERIMENTATION Numerical experiments are performed using the analysis area shown in Fig. 1. On average, the width of the coastal zone is about 300km and an idealized cyclone moves along the indicated north-westerly track for about 3 days before landfall at the coast of Andhra Pradesh. Thus, all conditions are identical to those prescribed in [1,2]. The idealized cyclonic wind field is simulated as in [2] by representing the azimuthal wind speed in the form Vo(r/R) /2
V=
for r ~
Voexp(R.~_£)forr>R,
(4.1)
166
B. JOHNSet al.
where Vo = 70 m s-1, R = 80 km and c = 240 km. The associated wind stress is calculated from a quadratic law with a friction coefficient of 2.8 x 10 -3. In this paper, we have taken a basic setting in which m = 10, n = 49, • = 0.04, ~0= 0.001 (resolution R1). Thus, ~,~ = 1.28 and At/-0.14 and corresponding discrete values of ~ and ,/are given in Table 1. From this, we note that the first off-shore grid-point at which the elevation is computed is, on average, about 6 km from the coastline. The estimated value of the elevation at a coastal station is derived by linear extrapolation from the elevation at the first off-shore point and that situated, on average, about 60 km from the coastline. In our earlier numerical simulations employing a uniform grid-spacing across the coastal zone the linear extrapolation to coastal stations was based upon elevations computed at grid-points respectively about 30 and 90 km from the coastline. The present technique therefore yields a much refined numerical scheme in which account may be taken of more detailed variations in the near-coastal bathymetry. The increased coastal resolution is obtained at the cost of a coarser resolution in the deep water adjacent to the eastern open-sea boundary. In this region, as shown later, variations in the bathymetry are of less importance than in shallow water although the resolution must still be adequate to resolve the scale of the forcing cyclonic wind pattern. Our object in the experimentation is to investigate the dependence of the surge response on variations in the bathymetric specification. For this purpose, it is convenient and, with an appropriate choice of parameters, reasonably realistic to represent the gross bathymetry in the Bay of Bengal by a function of the form y (y/L)'/
(4.2) In (4.2), a,/3 and 3' are disposable constants and ho is a depth scale that corresponds to the equilibrium depth of water along the coastline ~ = 0 and at y = L. Thus, values of a a n d / / m a y be selected to model the maximum depth of water at the south-eastern extremity of the analysis area (~ = l, y = 0) together with the off-shore slope of the sea-floor. The value of 3' may be chosen to model the northward rate of reduction in the depth of the water. Equation (4.2) is then applied to generate the bathymetry at the discrete q-points by using numerically inverted values of ~ obtained from (2.17).
Table 1. Discrete values of ~ and ~ with resolution RI. A~?= 0.14. nl~n
o
o
1
0.020
2
0.099
3
0.211
4
0.335
5
0.463
6
0.595
7
0. 729
8
0.864
9
! .000
On the effectof bathymetryin numericalstormsurgesimulationexperiments
167
(i) Comparison of non-uniform and uniform grid models In this experiment, we have used the grid-refinement model (M1) with a bathymetry B1 corresponding to a -- 4.6052,/3 = 0.25, 3' -- 0.5, ho = 5 m in (4.2) together with the resolution R1. This yields a fairly realistic approximation to the gross bathymetry in the immediate off-shore region along the east coast of India but it considerably underestimates the maximum depths actually attained-particularly those in the south-eastern sector of the analysis area. With B1, the maximum depth of water is 500 m at ~:= 1, y = 0. At ~ = 0.02, y = .)27(about 7.6 km off-shore of Kavali), h = 13.26 m, whilst at ~ = 1/9, y = Y27,the depth is about 69% greater and equal to 22.35 m. We compare results from M1 with those from a model M2 (obtained by putting e = 0 in (2.17)) which corresponds to the type of coastal zone model described in [1] in which the scaled off-shore grid increment is uniform across the coastal zone and equal to 1/9. Two different bathymetries are used in experiments with M2. The first is B 1 whils*,the second is B2 and corresponds to a = 3.92, /3 = 0.75, y = 1.0, ho = 10 m. B2 reproduces almost exactly the bathymetry used in [1] and shown in Fig. 3 of that paper. It implies a maximum depth of 500 m at ~:= 1, y = 0 but at ~:= 1/9, y = Y27, h = 16.39 m. This is about 27% less than that at the same position with B1 but is within 5% of the average depth of approximately 15.6 m obtained with B1 for 0<~:< 1/9. The model bathymetric sections off Kavali calculated from B 1 and B2 are shown in Fig. 2. In Fig. 3, we give the time-history of the sea-surface elevation at Kavali and Divi as determined from M1 with R1 and M2 when each of these has bathymetry B1. In comparison with M2, we note that M1 produces a higher peak surge at both stations. At Kavali, the peak surge response calculated from M1 is about 29% higher than that computed from M2. This indicates that the depth of water at the first off-shore grid-points is critical in determining the surge at coastal stations. In this connection, the adequacy of the near-coastal bathymetric resolution in M1 has been investigated by running MI(B1) with m = 12, n =49, ~0=0.001, = 0.04 (resolution R2). At y = Y27,we note that the first off-shore grid-point is approximately 4.7 km from the coastline and the corresponding equilibrium depth is about 11.87 m. With R2, the depth of water at the first off-shore point is therefore reduced by about 45% compared with R1. However, at t = 55 hr, we find that the surge response at Kavali is within 2% of that obtained with R1. We infer from this result that R1 leads to a satisfactory resolution of the near-coastal bathymetry; by moving the first off-shore grid-point nearer the coastline, there is no significant change in the computed surge response at Kavali.
0
5O I.UJ
\ \
Z
\
"r
1oo
\
uJ
\ \
,50
0
I
0.25
I
0.5
I
0.75
~
1.0
Fig. 2. Model bathymetric sections off Kavali. Continuous line is from B 1. Broken line is from B2.
168
B. JOHNSet al.
5.0
OlVl
KAVALI
40 30 2.0 L~J CE
2"s- ~.0 Z
z
0
O
"~ -~.o uJ w
-2.0 -3.0 -40 -5.0 -6.0 -7.0 -8.0
I
l
I
I
I
0
20
40
60
80
100
TIME IN HOURS
Fig. 3. Surge elevations at Kavali and Divi computed from MI(BI) (continuous line) with resolution RI and from M2(B1) (broken line).
We next compared responses obtained from MI(B1) using R1 with those from M2(B2). The time-histories of the sea-surface elevation at Kavali and Divi are given in Fig. 4. We note that both models produce almost the same peak response at Kavali. With the coarser (and uniform) coastal zone resolution in M2 this supports our view that it is necessary to adjust appropriately the bathymetry at the first off-shore grid-points. This adjustment takes the form of the use of notional reduced depths that are more in accord with the average depth for 0<~ ~< 1/9 in MI(B1). Whilst this procedure is not necessarily recommended as an alternative to using MI(BI), it presumably explains why the simulations given in [1] were in reasonable agreement with available data derived from reports of flooding along part of the coast of Andhra Pradesh. (ii) Convergence studies In order to estimate the effect of overall truncation error in the numerical scheme, we next ran MI(B1) with m = 14, n = 49, to = 0.001, ~ = 0.0226 (resolution R3) and give the corresponding discrete values of ~ and ~ in Table 2. With R3, the positions of the first off-shore grid-points coincide exactly with those resulting from the use of R1. Accordingly, a precise comparison may be made between directly computed elevations at ~ = 0.02 with two different coastal zone resolutions. For ~> 0.02, we note from Table 2 that R3 leads to a coastal zone resolution increased by between 65 and 80% compared with RI. We are therefore able to check that the forcing wind-stress field is being properly resolved in the model. In Fig. 5, we give the distribution of the maximum sea-surface elevation (peak surge envelope) together with its time of occurrence at ~ = 0.02, 0 < y < L. These are computed from MI(B1) with both R1 and R3. For (y/L)> 0.4, it is noteworthy that the peak surge propagates parallel to the coastline with a phase that generally becomes later with increasing y.
ELEVATION
,'~ 'TJ
~.~"
IN M E T R E S
o
I
I
I xl
I ¢D
E
o
ELEVATION IN METRES
L,'n
~
o~
r
E~"
x
~
o
o
0
o
0
0
I
I
I
I
I
I
I
o
~
o
o
o
o
o
I
I
I
J
[
~-
E
=t ]~
~g
8~
o
r-
o E. "I"1
~°~
('I)
~.~ II o
0
A ~I",
TIME
IN
HOURS
c,~
B. JOHNS et al.
170
Table 2. Discrete values of ~ and "Owith resolution R3, A'0 0.089. =
n/An o
o
1
0.020
2
0.079
3
0.153
4
0.232
5
0.315
6
O. 398
7
0.483
8
O. 568
9
0.654
I0
0.740
11
0.826
12
0.913
13
1.000 i
In Fig. 6, the instantaneous level of the sea-surface is shown off Kavali at t = 50 h and t = 100h as a function of the scaled off-shore distance ~. These two instants correspond respectively to positive and negative phases of the coastal surge and are calculated using a resolution R5 given by m = 30, n = 49, ~0= 0.001, e = 0.0015. We see that the surge response has the characteristics of a northward propagating coastatly-trapped wave as described, e.g. in [7]. For the positive phase, the off-shore e-folding scale is given by l = 0.35 and so (A~)~=0//= 0.057 and (2.21) is clearly satisfied. The extrapolated surge envelope at coastal stations shows that peak elevations exceeding 1 m are predicted over a north-south distance of almost 475 km. This envelope may be atypical for surges along the east coast of India and is comparable with that reported for the surge generated by Hurricane Carla in 1961 in the Gulf of Mexico. In this, surges exceeding 1 m were monitored from Texas to the coast of Louisiana--a distance of over 800 km [8]. Further to our suggestion that the surge response consists of a northward travelling coastally-trapped wave, we refer to work reported by Heaps [9] on the numerical simulation of externally generated surges in the North Sea. The inference here is that the surge response propagates southwards along the east coast of the British Isles in the form of a trapped Kelvin wave. In this, the confinement of the peak surge energy to the coastal regions is a consequence of the earth's rotation. In contrast with this, a northward propagation of Kelvin waves along the east coast of India is not possible. Hence, a northward propagating coastally-trapped wave in this region must necessarily be related to the existence of the steep off-shore slope of the sea-floor. The nature of the response therefore appears related to the northward propagation of a species of edge wave as studied by Reid[10]. We note from Fig. 5 that the increased resolution obtained with R3 does not significantly alter the peak surge elevation at ~ = 0.02 with regard to either its amplitude or phase. The root
On the effect of bathymetry in numerical storm surge simulation experiments
171
0 2O
z 60 80 ° z 0 --
.~ .i
/" -2 - -
/ /
/ -4 0
I
025
I
0.5
I
0 35
f
1.0
Fig. 6. Instantaneous elevation, ~, of sea-surface off Kavali. ~ 6 at t = 5 0 h ; . . . . . . . t = 100 K . . . . corresponds to bathymetric section off Kavali computed from BI.
I0~ at
mean square difference between computed elevations at ~ = 0.02 (0 ~
172
B. JOHNS et al.
5.0 I l I
4.0
3.0
I
\,
-
I i I I
DIVI
,, ;..//
2.0 uJ
~1.0 W
_
Z
z
\,,
,, ,,/,/q ./
t / /
,
\
0
t~ -1.0 .._1 UA
!
-2.0
_
! !
-30
~4.0
-5.0
-6.0
-7.0
I
20
I
z.,0
1
~"
60 TIME IN HOURS
80
I
~00
Fig. 7. Surge elevations at Kavali and Divi computed from M2(B2) (broken line) and M2(B3) (continuous
line).
5. C O N C L U S I O N S
We have investigated the effect of different finite-difference resolutions, and their relation to the bathymetric specification, in a numerical storm surge model applicable to the east coast of India. It is shown that the proper resolution of the immediate near-coastal bathymetry is critical in determining the storm-induced sea-surface elevation. We have described a method by means of which the necessary resolution can be obtained. This involves a near-coastal continuous grid compression. Experiments using this grid refinement model have also been performed to investigate the necessary resolution for the representation of the spatial scale of the forcing cyclone. The track followed by the forcing cyclone in this investigation has a substantial northward long-shore component prior to the landfall of the storm. As a result of the experiments, we suggest that the associated surge response is predominantly a result of the northward propagation of a coastally-trapped wave together with a contribution from local wind-stress forcing. The near-coastal local amplitude of the trapped wave is strongly dependent on the local bathymetry and coastline c o ~ a t i o n . The geographical position of the peak surge elevation is therefore determined by a complicated dynamical process. This appears to contrast with the case of the almost normal incidence of a storm track on a straight coastline where, we suggest,
On the effect of bathymetry in numerical storm surge simulation experiments
5.0
173
--
KAVALI
L,.0
I
3.0
-
2.0
-
1.0
U3 ttl t',,' uJ
0
z z o
- I .0
-2.0 -.I UJ
-3.0
-40 -50 -6.0 ~7.0
20
/.0 TIME
60 IN HOURS
80
100
Fig. 8. Surge elevations at Kavali and Divi computed from MI(BI) (continuous line) and MI(B4) (broken line).
the surge response is predominantly a consequence of local wind-stress forcing. In this case, there is hardly any contribution from an alongshore propagating coastally-trapped wave. We have also shown that the coastal surge response is independent of the bathymetry in the far off-shore deep water regions. This is partly explained by the fact that the off-shore distance of these regions greatly exceeds the off-shore e-folding scale of the coastally-trapped wave. Acknowledgement--A large part of this work was carried out with support from the U.K. Overseas Development Administration originally under the auspices of the Imperial College Delhi Committee. REFERENCES 1. B. Johns, S. K. Dube, U. C. Mohanty and P. C. Sinha, Numerical simulationof the surge generated by the 1977Andhra cyclone. Q. Z It. Met. Soc. 107, 919-934 (1981). 2. B. Johns, S. K. Dube, P. C. Sinha, U. C. Mohanty and A. D. Rao, The simulation of a continuously deforming laterial boundary in problems involving the shallow water equations. Computers and Fluids 10(2), 105-116(1982). 3. C. P. Jelesnianski, A numerical calculation of storm tides induced by a tropical storm impingingon a continental shelf. Moil Weath. Rev. 93, 343-358 (1965). 4. P. K. Das, M. C. Sinha and V. Balasubramanyam,Storm sngres in the Bay of Bengal. (2..£ It. Met. Soc. leO, 437-449 (1974). 5. B. Johns and M. A. Ali, The numerical modellingof storm surges in the Bay of Bengal: (2. J, R. Met. Soc. 106,1-18 (1980).
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