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The numerical simulation of storm surges along the Bangladesh coast
Dynamics of Atmospheres and Oceans, 9 (1985) 121-133
121
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
THE NUMERICAL SIMULATION OF STORM SURGES ALONG THE BANGLADESH COAST
S.K. DUBE, PC. SINHA and G.D. ROY * Centre for Atmospheric Sciences, Indian Institute of Technology, Hauz Khas, New Delhi - I 10016 (India)
(Received September 16, 1983; revised October 22, 1984; accepted March 1, 1985)
ABSTRACT Dube, S.K., Sinha, P.C. and Roy, G.D., 1985. The numerical simulation of storm surges along the Bangladesh coast. Dyn. Atmos. Oceans, 9: 121-133. Using a set of basic hydrodynamic equations governing the motion in a sea, a vertically-integrated coastal zone numerical model has been developed for the simulation of storm surges generated by the tropical storms striking the Bangladesh coast. The model is fully nonlinear and uses a conditionally stable semi-explicit finite difference scheme for solving the relevant equations. In this model the analysis area extends from 87’E to 93”E along the south coast of Bangladesh and there are three open-sea boundaries situated along 87”E, 93”E and 19”N. With this analysis area it is possible to record, on average, 24 h of the surge generating capacity of most of the severe cyclones before landfall at the Bangladesh coast. Numerical experiments are performed with the help of this model to simulate the surges generated by several severe cyclonic storms which struck the Bangladesh coast during the past 25 y. To compare the model predicted surges with the observed sea-surface elevations, three cyclonic storms have been chosen for which reliable observations regarding storm characteristics and the associated surges were available. The predicted peak sea-surface elevations compare well with the observed values at Chittagong port.
1. INTRODUCTION
In the present work, a vertically-integrated coastal zone numerical model has been developed for the simulation of storm surges along the Bangladesh coast. The noteworthy difference in the formulation of this model from those developed earlier for the region (Das, 1972; Flier1 and Robinson, 1972; Das et al., 1974; Johns and Ah, 1980; Henry and Murty, 1982; Murty and Henry, 1983), is the way in which the topography of the Bangladesh coast is * Present address: Department 0377-0265/85/$03.30
of Mathematics, Dhaka College, Dhaka, Bangladesh.
0 1985 Elsevier Science Publishers B.V.
122
represented. The treatment of the coastal boundary in this model involves a procedure leading to a realistic curvilinear representation of the natural shore-line. In storm surge work, curvilinear boundary representations have previously been considered by many workers including Reid et al. (1977) and Thacker (1979), however, our procedure is similar to that reported by Johns et al. (1981). Using this model, we have performed numerical experiments to simulate the surges generated by severe cyclonic storms which struck the Bangladesh o
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123 coast during the past 25 y. The results are, however, presented for three cyclones (November 1970, May 1961 and October 1960) for which reliable observational data on surges were available for comparison. The time history of these cyclones is shown in Fig. 1. The results of our numerical experiments are encouraging and suggest that given the forecast of a cyclone track and its intensity the model may be used for estimating the surges along the Bangladesh coast on a real time basis. 2. FORMULATION OF THE MODEL The sphericity of the Earth's surface is neglected and we use a system of rectangular Cartesian coordinates in which the origin, O, is within the equilibrium level of the sea-surface. Ox points towards the south, Oy points towards the east 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 northern coastal boundary (the south coast of Bangladesh) is situated at x = ba(y ) and a southern open-sea boundary is at x = b2(y). The western and the eastern open-sea boundaries are at y = 0 and y = L, respectively. This configuration is shown in Fig. 2. The basic hydrodynamic equations of continuity and m o m e n t u m in the vertically integrated form may then be given by H, + ( H u ) x + (Ho)y = 0 (1) u, + UUx + vu, - h
1 { Fs _ Cfpu(u 2 +
= -gL +
1
0t'-[= nox =I-VVy+fU = --~)~, + ~ { G
s - Cfpv(u 2-t-
P2)1/2}
(2)
lP2)1/2}
(3)
where jr denotes the Coriolis parameter, the pressure is taken as hydrostatic
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,~
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.......
9(~E
, ......
9~E
. .......
9~E
,~ ,,
93 E
9zE
Fig. 2. The analysis area. - . . . . . : Open-sea boundaries O: 5-hourly position of the centre of cyclones.
124
and H is the total depth ~ + h. ( ~ , Gs) denotes the applied surface windstress and the bottom stress is parameterized in terms of a quadratic law. p denotes the water density and the friction coefficient, Cf, is taken as 2.6 × 10 -3. After Johns et al. (1982), the appropriate boundary conditions are given by
u - vbl v = 0
at x = b l ( y )
(4)
at x = b2(y )
(5)
v+(g/h)l/2~=O
at y = O
(6)
v -(g/h)l/2~ = 0
at y = L
(7)
U -- Ub2y -- ( g / h ) l / 2 ~
= 0
3. COORDINATE TRANSFORMATION
To facilitate the numerical treatment of an irregular boundary configuration we introduce a coordinate transformation, similar to that used by Johns et al. (1981), which is based upon a new set of independent variables ~, y and t, where ~= {X-bl(y)}/b(y
),b(y)=b2(y)-b,(y
)
(8)
This mapping which is non-conformal, transforms the analysis area into a rectangular computational domain given by 0 ~ ~ ~ 1, 0 ~ y ~< L. Taking ~, y and t as new independent coordinates, (1)-(3) may be transformed to
(b~), + ( b H U ) ~ + ~v = 0
(9)
= -gH~,+b~/o-Cf~(u2+
fi,+(Vfi)~+(vfi)y-fb ~, + (U~)~ + (V~)y + f f t =
-gH{b[v-(bly
vz)~/2/H
(10)
+ ~by)~}
+ b G , / o - clg,(u 2 + v2)I/
/H
(11)
where
bU = u - ( b,y + ,~b~)v
(12)
m-/(u, v)
(13)
The boundary conditions (4) and (5) reduce to U = 0
at ~ = 0
(14)
bU-(g/h)l/2~=O
at~=l
(15)
Equations (9)-(11) form the basic set for the numerical solution process.
125 4. NUMERICAL PROCEDURE For the solution of the eqs. (9)-(11) subject to the relevant boundary conditions, a conditionally stable explicit finite-difference scheme with a staggered grid is used. The details of the discretization procedure and the finite-difference grid selection are completely analogous to those reported in Johns et al. (1981, 1982). The stability characteristics of the computational scheme has also been discussed in details by Johns et al. (1981). In fact stability is only conditional upon the time step being limited by the space increment and the gravity wave speed. 5. NUMERICAL EXPERIMENTS In this model the analysis area extends from 87°E to 93°E along the south coast of Bangladesh and there are three open-sea boundaries situated along 87°E, 93°E and 19°N (Fig. 2). Thus, on an average, the width of the coastal zone is about 270 km while the east-west extent, L, of the analysis area is about 600 km. The idealised, but representative, bathymetry used in the model is indicated in Fig. 3. The numerical experiments are performed with the help of this model to simulate the surges generated by several severe cyclonic storms which struck the Bangladesh coast during the past 25 y. However, the results are presented only for three cyclonic storms (October 1960, May 1961, November 1970) for which reliable observational data was available for comparison. An idealised wind field associated with these cyclones is simulated by applying empirically-based formula (Jelesnianski, 1965)
V=
Vo(r/R) /2
for
r ~
Vo(R/r)~/2
for
r>R
(16)
where V0 is the maximum sustained wind, R, the radius of maximum wind and r is the distance from the centre o f the cyclone. Using reports from the annual summary of the Indian Weather Review (published by the India Meteorological Department) and the information given by Das et al. (1974), the following values of V0 and R have been taken (1) 8-13 November, 1970 cyclone: (2) 5'-9 May, 1961 cyclone: (3) 8-11 October, 1960 cyclone:
V0 = 50 ms -1, V0 = 45 ms -1, V0 = 45 ms -a,
R = 40 km. R = 40 km. R = 40 km.
In each of the experiments, we prescribe that the cyclone moves along the idealised straight-line track shown in Fig. 2 with uniform speed of translation. The five hourly positions of the centres of the cyclones are also indicated in the figure.
126 • C HANDPE~R
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Fig. 3. Idealised bathymetry (in metres) used in the model.
The optimum finite-difference grid in the computational plane is chosen with m = 20 and n = 21. This is obtained as a result of several experiments performed with the 1970 Chittagong cyclone for testing the adequacy of the resolution. With our choice of open-sea boundaries, the north-south space increment varies between 8 km and 20.5 km and Ay = 30 km. In our numerical experiments, we prescribed an initial state of rest and integrated the governing equations ahead in time for upto a total of 50 h. A time-step of 3 min was found to be consistent with the computational stability. We also considered the sea-surface response during several hours after landfall when the system starts to enter the resurgence phase. 6. RESULTS A N D DISCUSSIONS
The surges associated with the 1970 Chittagong cyclone have been computed by using a number of off-shore grid resolutions without altering the size of the basin. However, the results are presented only for two resolutions (m = 10, n = 21 and rn = 20, n = 21) for comparison and for an optimum selection of the grid-size. In Fig. 4 we give the distribution of the predicted maximum sea-surface
127 7-0,
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Fig. 4. Peak surge envelope and its time of occurrence along Bangladesh coast (for November 1970 cyclone). • .... • Peak surge envelope for m = 10. - Peak surge envelope for m = 20. • . . . . . -e Time of occurrence for m = 10. - . . . . . Time of occurrence for m = 20. * ..... * (Range of observed sea-surface elevation minus the predicted astronomical tide.) • Time of landfall. § Point of landfall.
e l e v a t i o n ( p e a k surge e n v e l o p e ) , its time o f o c c u r r e n c e a n d the o b s e r v e d surge a l o n g the B a n g l a d e s h coast. It m a y be s e e n f r o m the figure that b y i n c r e a s i n g m f r o m 10 to 20 the p e a k surge at Maijdi a n d C h i t t a g o n g increases b y a b o u t 11% a n d 26%, respectively. Further, the p e a k surge at Maijdi o c c u r s a l m o s t at the s a m e time for the t w o r e s o l u t i o n s w h i l e at C h i t t a g o n g the p e a k o c c u r s a b o u t 1 h earlier for the higher r e s o l u t i o n case. A further i n c r e a s e in the value o f m d o e s n o t i n d i c a t e a n y significant c h a n g e either in the a m p l i t u d e or the p h a s e o f the p e a k surge at these stations. T h e a d e q u a c y o f the e a s t - w e s t r e s o l u t i o n has also b e e n tested by increas-
128 ing the value of n, which has not resulted in any significant change in the elevations and their time of occurrence along the coastal stations. These tests on the resolutions indicate that a 20 × 21 grid is the optimum resolution for this model. Figure 4 provides an idea of the coastal stretch at which the significant surge may be expected. It may be seen from the figure that the maximum surge height of 5.5 m is predicted at Maijdi which is about 40 km to the left of the landfall point. A maximum surge of about 4.9 m is predicted at Chittagong Port (about 28 km to the right of the landfall point) at 0520 h BST on N o v e m b e r 13, 1970. A peak elevation of 6 - 9 m above mean sea-level was estimated at this station on the morning of 13 November while according to the tide table the predicted astronomical tide was 1.8 m at about the same time. Thus, the estimated sea-surface elevation was in excess of 4.2-7.2 m over the usual high tide. Hence, it may be seen that our predicted surge (4.9 m) is in good agreement with the observed range of maximum surge (4.2-7.2 m) at Chittagong. 5"5 5"0
/'~4 BHOLAISLAND
~H
~.o
/ /
MAIJDI
\ t~ ~ ,,
\
3'0 z
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Fig. 5. Time variation of the predicted sea-surface elevation at coastal stations. The symbol • on the time-scale indicates the time of landfall.
129
Figure 5 depicts the time variation of the storm surge at four stations along the Bangladesh coast. All the three stages of the surge-forerunner, the main surge, and the resurgence phase may clearly be seen from the temporal variation of the coastal surface variations. The maximum surge height of 5.5 m is predicted at Maijdi. The peak occurred at 0245 h BST on the morning of 13 November. This main surge remains for a very short period (less than 30 min) and then enters the resurgence phase. During this phase the sea-surface elevation falls rapidly from its maximum value and becomes negative after about the next 5 h period and a sea-surface depression of about 2.3 m is predicted at 1530 h BST on 13 November. The predicted time history of the surge at Chittagong indicates a maximum sea-surface elevation of 4.9 m in the morning of 13 November at about 0520 h BST which is in close agreement with the time of landfall. This main surge persists for about 30 min before falling rapidly and becoming negative at 1245 h BST. At Chittagong, a maximum sea-surface depression of about 60 cm is predicted at 1630 h BST on 13 November with a subsequent increase in the water height leading to a depression of about 17 cm in the early morning of 14 November.
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_
, eMAIjD I
I
~ C H I T T A G O N
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N~'NO~~ R
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~_ .
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~
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o o
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/
i
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'-, ...............
,7
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8~ E
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.........
t.~ ...............
90 E
J~ ..............
9, E
j ..................
9~E
l
9~E
Fig. 6. Contours (in metres) of equal sea-surface elevation at 0520 h BST on 13 November.
130
The predicted surge height at Bhola Island (about 110 km to the west of landfall) is about 5 m which occurs at 0000 h BST on 13 November. Here again the main surge is short-lived (less than 30 min) and the surge becomes negative within 5 h of the occurrence of its peak value. A sea-surface depression of 2.3 m is predicted at 0945 h BST on 13 November. At Cox's Bazar which is about 140 km to the SE of landfall, the sea-surface elevation rises gradually to attain its peak value of 2.8 m at 0445 h BST on 13 November. This high sea-surface elevation is again very short-lived and it falls rapidly to become negative in the afternoon of 13 November. A negative surge of about 16 cm is predicted at 1700 h BST after which the elevation changes in the form of a damped free oscillation. In Fig. 6 contours of equal sea-surface elevation are given for the coastal
60
3~0
2~0
~0
~
~00
200
500
50
!
~o0o I oo~ tn
/
z
4.0
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t / ¢
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~ ~ ~
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Fig. 7. Peak surge e n v e l o p e a n d its time of o c c u r r e n c e a l o n g a h y p o t h e t i c a l straight coast-line. - Peak surge e n v e l o p e . - . . . . . T i m e of occurrence. • T i m e of landfall. § Point of landfall.
131
regions of Bangladesh. This spatial distribution of the surge corresponds to about the time of landfall, that is, at 0520 h BST. It may be seen from the figure that a positive surge of more than 4.5 m occurs near Chittagong. The surge is positive everywhere on the eastern side of the basin and negative on the western side. It may be noted from the above discussions that the highest surge occurs at Maijdi which is to the left of the landfall and is contrary to the usual thinking that the maximum surge should appear to the right of the landfall point. This may be a point of contention which should be resolved. We attribute the occurrence of the peak surge towards left of the landfall to the local coastal geometry of the region. To confirm this surge heights associated with the 1970 Chittagong cyclone were computed by considering a straightline coast at the place of landfall. Figure 7 gives the maximum predicted surge along this straight coast. It may be seen that the maximum surge now occurs at Chittagong port, instead of Maijdi, which is to the right of the
2~0
1~0
0'
' 100
' 200
3~ 0
' 400 Kid
4.(
3.(
/I
E
",
/ z
l ,
;
ss"
2.0 Ix
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m
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s"
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-R
~: o
--
c~
~
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surge envelope and its time of occurrence along Bangladesh coast (for May 5 - 9 , - P e a k surge envelope. - . . . . . Time of occurrence. * (Observed sea-surface elevation minus the predicted astronomical tide). • Time of landfall. § Point of landfall. F i g . 8. P e a k
1961
cyclone).
132
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i
'
i
~00
3OO
200
KM
6'0
1 1900 ]
50
I
o~
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1400 i J
'2_
/ z
,
3-0
/
0800~
== 2-0
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00~,
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i
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ct:
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Fig. 9. As Fig. 8 except computed for October 8-11, 1960 cyclone.
landfall point. This result is consistent with the earlier findings of Dube et al. (1982) that the location of the peak surge depends on the local coastal geometry of the place of landfall. Figures 8 and 9 give the distribution of predicted maximum surge, its time of occurrence and the observed surge along the Bangladesh coast for May 5-9, 1961 and October 8-11, 1960 cyclones, respectively. It may be seen from the figures that the predicted maximum sea-surface elevations at Chittagong are in good agreement with the observed values. 7. C O N C L U S I O N S
A coastal zone numerical model has been developed for simulating the surges generated by severe cyclonic storms striking the Bangladesh coast.
133 T h e following general conclusions m a y b e d r a w n o n the basis o f the a b o v e results: (1) the m o d e l p r e d i c t e d surge c o m p a r e s well with the actual o b s e r v a t i o n s ; and (2) the m o d e l can b e b a s e d for simulating the surge g e n e r a t e d b y a c y c l o n e o f a n y speed or intensity. It m a y also b e used for p r e d i c t i n g the surges o n a n o p e r a t i o n a l basis. REFERENCES Das, P.K., 1972. Prediction model for storm surges in the Bay of Bengal. Nature, 239: 211-213. Das, P.K., Sinha, M.C. and Balasubramanyam, V., 1974. Storm surges in the Bay of Bengal. Q. J.R. Meteorol. Soc., 100: 437-449. Dube, S.K., Sinha, P.C. and Rao, A.D., 1982. The effect of coastal geometry on the location of peak surge. Mausam, 33: 445-450. Flierl, G.R. and Robinson, A.R., 1972. Deadly surges in the Bay of Bengal: dynamics and storm-fide tables. Nature, 239: 213-215. Henry, R.F. and Murty, T.S., 1982. Tides in the Bay of Bengal. In: G.A. Keramidas and C.A. Brebbia (Editors), Computational Methods and Experimental Measurements. SpringerVerlag, New York, pp. 541-550. Jelesnianski, C.P., 1965. A numerical calculation of storm tides induced by a tropical storm impinging on a continental shelf. Mon. Weather Rev., 93: 343-358. Johns, B. and Ali, A., 1980. The numerical modelling of storm surges in the Bay of Bengal. Q. J.R. Meteorol. Soc., 106: 1-18. Johns, B., Dube, S.K., Mohanty, U.C. and Sinha, P.C., 1981. Numerical simulation of the surge generated by the 1977 Andhra cyclone. Q. J.R. Meteorol. Soc., 107: 919-934. Johns, B., Dube, S.K., Sinha, P.C., Mohanty, U.C. and Rao, A.D., 1982. The simulation of a continuously deforming lateral boundary in problems involving the shallow water equations. Computers and Fluids, 10: 105-116. Murty, T.S. and Henry, R.F., 1983. Tides in the Bay of Bengal. J. Geophys. Res., 88: 6069-6076. Reid, R.O., Vastano, A.C., Whitaker, R.E. and Wanstrath, J.J., 1977. Experiments on storm surge simulation. In: Goldberg et al. (Editors), The Sea. Wiley-Interscience Publication, New York, London, Sydney, Toronto, 6: 145-168. Thacker, W.C., 1979. Irregular grid finite difference techniques for storm surge calculations for curving coast lines. In: J.C.J. Nihoul (Editor), Marine Forecasting. Elsevier Ocean Ser., pp. 261-283.
121
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
THE NUMERICAL SIMULATION OF STORM SURGES ALONG THE BANGLADESH COAST
S.K. DUBE, PC. SINHA and G.D. ROY * Centre for Atmospheric Sciences, Indian Institute of Technology, Hauz Khas, New Delhi - I 10016 (India)
(Received September 16, 1983; revised October 22, 1984; accepted March 1, 1985)
ABSTRACT Dube, S.K., Sinha, P.C. and Roy, G.D., 1985. The numerical simulation of storm surges along the Bangladesh coast. Dyn. Atmos. Oceans, 9: 121-133. Using a set of basic hydrodynamic equations governing the motion in a sea, a vertically-integrated coastal zone numerical model has been developed for the simulation of storm surges generated by the tropical storms striking the Bangladesh coast. The model is fully nonlinear and uses a conditionally stable semi-explicit finite difference scheme for solving the relevant equations. In this model the analysis area extends from 87’E to 93”E along the south coast of Bangladesh and there are three open-sea boundaries situated along 87”E, 93”E and 19”N. With this analysis area it is possible to record, on average, 24 h of the surge generating capacity of most of the severe cyclones before landfall at the Bangladesh coast. Numerical experiments are performed with the help of this model to simulate the surges generated by several severe cyclonic storms which struck the Bangladesh coast during the past 25 y. To compare the model predicted surges with the observed sea-surface elevations, three cyclonic storms have been chosen for which reliable observations regarding storm characteristics and the associated surges were available. The predicted peak sea-surface elevations compare well with the observed values at Chittagong port.
1. INTRODUCTION
In the present work, a vertically-integrated coastal zone numerical model has been developed for the simulation of storm surges along the Bangladesh coast. The noteworthy difference in the formulation of this model from those developed earlier for the region (Das, 1972; Flier1 and Robinson, 1972; Das et al., 1974; Johns and Ah, 1980; Henry and Murty, 1982; Murty and Henry, 1983), is the way in which the topography of the Bangladesh coast is * Present address: Department 0377-0265/85/$03.30
of Mathematics, Dhaka College, Dhaka, Bangladesh.
0 1985 Elsevier Science Publishers B.V.
122
represented. The treatment of the coastal boundary in this model involves a procedure leading to a realistic curvilinear representation of the natural shore-line. In storm surge work, curvilinear boundary representations have previously been considered by many workers including Reid et al. (1977) and Thacker (1979), however, our procedure is similar to that reported by Johns et al. (1981). Using this model, we have performed numerical experiments to simulate the surges generated by severe cyclonic storms which struck the Bangladesh o
.
27N I
.
.
.
.
.
•
--
]
!
BANGLA
i
i I
24 ° L-
DESH
- "~
INDIA CACUTTA
,
:HI T~-,~:)NG
BALASORE v
21°L
BURMA
BHUBANESHWAR
.% ".
O -®
Il [
i BAY OF BENGAL
i
i
I i I
i i
DEPRESSION
,~
.....
SrOR.
)~. /.%
:E
t
L
84°E
I
. . . . . . . . . . . . . . .
87 °
Fig. 1. Actual track of the cyclones.
1-
90°
. . . . . . . . . . . . . .
L o-
93
o
96
123 coast during the past 25 y. The results are, however, presented for three cyclones (November 1970, May 1961 and October 1960) for which reliable observational data on surges were available for comparison. The time history of these cyclones is shown in Fig. 1. The results of our numerical experiments are encouraging and suggest that given the forecast of a cyclone track and its intensity the model may be used for estimating the surges along the Bangladesh coast on a real time basis. 2. FORMULATION OF THE MODEL The sphericity of the Earth's surface is neglected and we use a system of rectangular Cartesian coordinates in which the origin, O, is within the equilibrium level of the sea-surface. Ox points towards the south, Oy points towards the east 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 northern coastal boundary (the south coast of Bangladesh) is situated at x = ba(y ) and a southern open-sea boundary is at x = b2(y). The western and the eastern open-sea boundaries are at y = 0 and y = L, respectively. This configuration is shown in Fig. 2. The basic hydrodynamic equations of continuity and m o m e n t u m in the vertically integrated form may then be given by H, + ( H u ) x + (Ho)y = 0 (1) u, + UUx + vu, - h
1 { Fs _ Cfpu(u 2 +
= -gL +
1
0t'-[= nox =I-VVy+fU = --~)~, + ~ { G
s - Cfpv(u 2-t-
P2)1/2}
(2)
lP2)1/2}
(3)
where jr denotes the Coriolis parameter, the pressure is taken as hydrostatic
BANGLADESH )i MR,JOt I
23°n
/
CALCUTTA j 3HO ~,.,~..~--------../'~CHITTAGONG y " CHALNA IS AN~ ~ 22°n
%. x/
INDIA
,_
/
,CONTAI /
/J
"~
" 'P
I
zfn
!
zcn
/
PAR~,P.,"
• ./" PURl ~,-" /. .
85°E
86E
~
~'~
',F
X
~ Z;
/~.
,~
/x,~
~.AKVAB ',LJ-\
~,-.....
8?°E
J . . . . . .
88°E
, ....
8~E
,__,
.......
9(~E
, ......
9~E
. .......
9~E
,~ ,,
93 E
9zE
Fig. 2. The analysis area. - . . . . . : Open-sea boundaries O: 5-hourly position of the centre of cyclones.
124
and H is the total depth ~ + h. ( ~ , Gs) denotes the applied surface windstress and the bottom stress is parameterized in terms of a quadratic law. p denotes the water density and the friction coefficient, Cf, is taken as 2.6 × 10 -3. After Johns et al. (1982), the appropriate boundary conditions are given by
u - vbl v = 0
at x = b l ( y )
(4)
at x = b2(y )
(5)
v+(g/h)l/2~=O
at y = O
(6)
v -(g/h)l/2~ = 0
at y = L
(7)
U -- Ub2y -- ( g / h ) l / 2 ~
= 0
3. COORDINATE TRANSFORMATION
To facilitate the numerical treatment of an irregular boundary configuration we introduce a coordinate transformation, similar to that used by Johns et al. (1981), which is based upon a new set of independent variables ~, y and t, where ~= {X-bl(y)}/b(y
),b(y)=b2(y)-b,(y
)
(8)
This mapping which is non-conformal, transforms the analysis area into a rectangular computational domain given by 0 ~ ~ ~ 1, 0 ~ y ~< L. Taking ~, y and t as new independent coordinates, (1)-(3) may be transformed to
(b~), + ( b H U ) ~ + ~v = 0
(9)
= -gH~,+b~/o-Cf~(u2+
fi,+(Vfi)~+(vfi)y-fb ~, + (U~)~ + (V~)y + f f t =
-gH{b[v-(bly
vz)~/2/H
(10)
+ ~by)~}
+ b G , / o - clg,(u 2 + v2)I/
/H
(11)
where
bU = u - ( b,y + ,~b~)v
(12)
m-/(u, v)
(13)
The boundary conditions (4) and (5) reduce to U = 0
at ~ = 0
(14)
bU-(g/h)l/2~=O
at~=l
(15)
Equations (9)-(11) form the basic set for the numerical solution process.
125 4. NUMERICAL PROCEDURE For the solution of the eqs. (9)-(11) subject to the relevant boundary conditions, a conditionally stable explicit finite-difference scheme with a staggered grid is used. The details of the discretization procedure and the finite-difference grid selection are completely analogous to those reported in Johns et al. (1981, 1982). The stability characteristics of the computational scheme has also been discussed in details by Johns et al. (1981). In fact stability is only conditional upon the time step being limited by the space increment and the gravity wave speed. 5. NUMERICAL EXPERIMENTS In this model the analysis area extends from 87°E to 93°E along the south coast of Bangladesh and there are three open-sea boundaries situated along 87°E, 93°E and 19°N (Fig. 2). Thus, on an average, the width of the coastal zone is about 270 km while the east-west extent, L, of the analysis area is about 600 km. The idealised, but representative, bathymetry used in the model is indicated in Fig. 3. The numerical experiments are performed with the help of this model to simulate the surges generated by several severe cyclonic storms which struck the Bangladesh coast during the past 25 y. However, the results are presented only for three cyclonic storms (October 1960, May 1961, November 1970) for which reliable observational data was available for comparison. An idealised wind field associated with these cyclones is simulated by applying empirically-based formula (Jelesnianski, 1965)
V=
Vo(r/R) /2
for
r ~
Vo(R/r)~/2
for
r>R
(16)
where V0 is the maximum sustained wind, R, the radius of maximum wind and r is the distance from the centre o f the cyclone. Using reports from the annual summary of the Indian Weather Review (published by the India Meteorological Department) and the information given by Das et al. (1974), the following values of V0 and R have been taken (1) 8-13 November, 1970 cyclone: (2) 5'-9 May, 1961 cyclone: (3) 8-11 October, 1960 cyclone:
V0 = 50 ms -1, V0 = 45 ms -1, V0 = 45 ms -a,
R = 40 km. R = 40 km. R = 40 km.
In each of the experiments, we prescribe that the cyclone moves along the idealised straight-line track shown in Fig. 2 with uniform speed of translation. The five hourly positions of the centres of the cyclones are also indicated in the figure.
126 • C HANDPE~R
--
i
,
--
CALCUTTA
-KHULNA
I
•
•
" 2~N
-
•
BARISAL ~
BHOLA. ~
CONTAI
MAIJDf
T ~ e C H I T TAGONG
\
ISLj~
10
ocox's ~AR
2~°N
87E
8BE
8~E/-~O0
90E
91 E
92 E
93 E
Fig. 3. Idealised bathymetry (in metres) used in the model.
The optimum finite-difference grid in the computational plane is chosen with m = 20 and n = 21. This is obtained as a result of several experiments performed with the 1970 Chittagong cyclone for testing the adequacy of the resolution. With our choice of open-sea boundaries, the north-south space increment varies between 8 km and 20.5 km and Ay = 30 km. In our numerical experiments, we prescribed an initial state of rest and integrated the governing equations ahead in time for upto a total of 50 h. A time-step of 3 min was found to be consistent with the computational stability. We also considered the sea-surface response during several hours after landfall when the system starts to enter the resurgence phase. 6. RESULTS A N D DISCUSSIONS
The surges associated with the 1970 Chittagong cyclone have been computed by using a number of off-shore grid resolutions without altering the size of the basin. However, the results are presented only for two resolutions (m = 10, n = 21 and rn = 20, n = 21) for comparison and for an optimum selection of the grid-size. In Fig. 4 we give the distribution of the predicted maximum sea-surface
127 7-0,
'
'
300
100
KM
6.0
5.0
1I
II
/;'
"%%
4.C
~1 II
'x
w
~
I I
!
' z -
g
;7
3-(:
w
7"
2-C
0
1.C
BANGLADESH , ~£
o-or--" <
WEST
~ < in
___ < I
COAST , EAST' ~ P
•
'~
u
Fig. 4. Peak surge envelope and its time of occurrence along Bangladesh coast (for November 1970 cyclone). • .... • Peak surge envelope for m = 10. - Peak surge envelope for m = 20. • . . . . . -e Time of occurrence for m = 10. - . . . . . Time of occurrence for m = 20. * ..... * (Range of observed sea-surface elevation minus the predicted astronomical tide.) • Time of landfall. § Point of landfall.
e l e v a t i o n ( p e a k surge e n v e l o p e ) , its time o f o c c u r r e n c e a n d the o b s e r v e d surge a l o n g the B a n g l a d e s h coast. It m a y be s e e n f r o m the figure that b y i n c r e a s i n g m f r o m 10 to 20 the p e a k surge at Maijdi a n d C h i t t a g o n g increases b y a b o u t 11% a n d 26%, respectively. Further, the p e a k surge at Maijdi o c c u r s a l m o s t at the s a m e time for the t w o r e s o l u t i o n s w h i l e at C h i t t a g o n g the p e a k o c c u r s a b o u t 1 h earlier for the higher r e s o l u t i o n case. A further i n c r e a s e in the value o f m d o e s n o t i n d i c a t e a n y significant c h a n g e either in the a m p l i t u d e or the p h a s e o f the p e a k surge at these stations. T h e a d e q u a c y o f the e a s t - w e s t r e s o l u t i o n has also b e e n tested by increas-
128 ing the value of n, which has not resulted in any significant change in the elevations and their time of occurrence along the coastal stations. These tests on the resolutions indicate that a 20 × 21 grid is the optimum resolution for this model. Figure 4 provides an idea of the coastal stretch at which the significant surge may be expected. It may be seen from the figure that the maximum surge height of 5.5 m is predicted at Maijdi which is about 40 km to the left of the landfall point. A maximum surge of about 4.9 m is predicted at Chittagong Port (about 28 km to the right of the landfall point) at 0520 h BST on N o v e m b e r 13, 1970. A peak elevation of 6 - 9 m above mean sea-level was estimated at this station on the morning of 13 November while according to the tide table the predicted astronomical tide was 1.8 m at about the same time. Thus, the estimated sea-surface elevation was in excess of 4.2-7.2 m over the usual high tide. Hence, it may be seen that our predicted surge (4.9 m) is in good agreement with the observed range of maximum surge (4.2-7.2 m) at Chittagong. 5"5 5"0
/'~4 BHOLAISLAND
~H
~.o
/ /
MAIJDI
\ t~ ~ ,,
\
3'0 z
'S 8AJAR
2C
[
¢,
,
"%. . . . .
,
8sT
.
0"¢ _
~_
NOV 12,1970
~
\
\
\
\NO¥
~ - - -
13,1970
/
/
NOV lt,, 1970
-2-C
-30
Fig. 5. Time variation of the predicted sea-surface elevation at coastal stations. The symbol • on the time-scale indicates the time of landfall.
129
Figure 5 depicts the time variation of the storm surge at four stations along the Bangladesh coast. All the three stages of the surge-forerunner, the main surge, and the resurgence phase may clearly be seen from the temporal variation of the coastal surface variations. The maximum surge height of 5.5 m is predicted at Maijdi. The peak occurred at 0245 h BST on the morning of 13 November. This main surge remains for a very short period (less than 30 min) and then enters the resurgence phase. During this phase the sea-surface elevation falls rapidly from its maximum value and becomes negative after about the next 5 h period and a sea-surface depression of about 2.3 m is predicted at 1530 h BST on 13 November. The predicted time history of the surge at Chittagong indicates a maximum sea-surface elevation of 4.9 m in the morning of 13 November at about 0520 h BST which is in close agreement with the time of landfall. This main surge persists for about 30 min before falling rapidly and becoming negative at 1245 h BST. At Chittagong, a maximum sea-surface depression of about 60 cm is predicted at 1630 h BST on 13 November with a subsequent increase in the water height leading to a depression of about 17 cm in the early morning of 14 November.
•
I
KHULNA CALCUTTA
•
..
•
_
, eMAIjD I
I
~ C H I T T A G O N
G
N~'NO~~ R
-I. 0
ZORN'-
°
BARISAL
BHOLA
--2"
~_ .
--"~---'~-
19
~
CHANDPUR
o o
.
/
i
! N
'-, ...............
,7
E
.t; ...............
8~ E
Ji,
89 ~
.........
t.~ ...............
90 E
J~ ..............
9, E
j ..................
9~E
l
9~E
Fig. 6. Contours (in metres) of equal sea-surface elevation at 0520 h BST on 13 November.
130
The predicted surge height at Bhola Island (about 110 km to the west of landfall) is about 5 m which occurs at 0000 h BST on 13 November. Here again the main surge is short-lived (less than 30 min) and the surge becomes negative within 5 h of the occurrence of its peak value. A sea-surface depression of 2.3 m is predicted at 0945 h BST on 13 November. At Cox's Bazar which is about 140 km to the SE of landfall, the sea-surface elevation rises gradually to attain its peak value of 2.8 m at 0445 h BST on 13 November. This high sea-surface elevation is again very short-lived and it falls rapidly to become negative in the afternoon of 13 November. A negative surge of about 16 cm is predicted at 1700 h BST after which the elevation changes in the form of a damped free oscillation. In Fig. 6 contours of equal sea-surface elevation are given for the coastal
60
3~0
2~0
~0
~
~00
200
500
50
!
~o0o I oo~ tn
/
z
4.0
3-0
3500 I
t / ¢
300~
,'" "'"
o
Tgo0
20
72 1.0
"
BANGLADESH 0-0
WEST
I
I
~< -
~ ~ ~
1400
!
COAST I
o0
~
EAST
~
I11
Fig. 7. Peak surge e n v e l o p e a n d its time of o c c u r r e n c e a l o n g a h y p o t h e t i c a l straight coast-line. - Peak surge e n v e l o p e . - . . . . . T i m e of occurrence. • T i m e of landfall. § Point of landfall.
131
regions of Bangladesh. This spatial distribution of the surge corresponds to about the time of landfall, that is, at 0520 h BST. It may be seen from the figure that a positive surge of more than 4.5 m occurs near Chittagong. The surge is positive everywhere on the eastern side of the basin and negative on the western side. It may be noted from the above discussions that the highest surge occurs at Maijdi which is to the left of the landfall and is contrary to the usual thinking that the maximum surge should appear to the right of the landfall point. This may be a point of contention which should be resolved. We attribute the occurrence of the peak surge towards left of the landfall to the local coastal geometry of the region. To confirm this surge heights associated with the 1970 Chittagong cyclone were computed by considering a straightline coast at the place of landfall. Figure 7 gives the maximum predicted surge along this straight coast. It may be seen that the maximum surge now occurs at Chittagong port, instead of Maijdi, which is to the right of the
2~0
1~0
0'
' 100
' 200
3~ 0
' 400 Kid
4.(
3.(
/I
E
",
/ z
l ,
;
ss"
2.0 Ix
i
m
i i
m
.J
1-0 i
s"
-I BANGLADESH
COAST
0"0 tY tt
WEST
~
-R
~: o
--
c~
~
EAST
1m
surge envelope and its time of occurrence along Bangladesh coast (for May 5 - 9 , - P e a k surge envelope. - . . . . . Time of occurrence. * (Observed sea-surface elevation minus the predicted astronomical tide). • Time of landfall. § Point of landfall. F i g . 8. P e a k
1961
cyclone).
132
2;o
i
'
i
~00
3OO
200
KM
6'0
1 1900 ]
50
I
o~
40
1400 i J
'2_
/ z
,
3-0
/
0800~
== 2-0
04001
!
10
00~,
2300
O
i
"'L"/
O0 WEST
-
,_
~
,
,
El z -~
~ o
L'} z 0
ct:
~.-
~
< lID
~
--18oo EAST
p
u
Fig. 9. As Fig. 8 except computed for October 8-11, 1960 cyclone.
landfall point. This result is consistent with the earlier findings of Dube et al. (1982) that the location of the peak surge depends on the local coastal geometry of the place of landfall. Figures 8 and 9 give the distribution of predicted maximum surge, its time of occurrence and the observed surge along the Bangladesh coast for May 5-9, 1961 and October 8-11, 1960 cyclones, respectively. It may be seen from the figures that the predicted maximum sea-surface elevations at Chittagong are in good agreement with the observed values. 7. C O N C L U S I O N S
A coastal zone numerical model has been developed for simulating the surges generated by severe cyclonic storms striking the Bangladesh coast.
133 T h e following general conclusions m a y b e d r a w n o n the basis o f the a b o v e results: (1) the m o d e l p r e d i c t e d surge c o m p a r e s well with the actual o b s e r v a t i o n s ; and (2) the m o d e l can b e b a s e d for simulating the surge g e n e r a t e d b y a c y c l o n e o f a n y speed or intensity. It m a y also b e used for p r e d i c t i n g the surges o n a n o p e r a t i o n a l basis. REFERENCES Das, P.K., 1972. Prediction model for storm surges in the Bay of Bengal. Nature, 239: 211-213. Das, P.K., Sinha, M.C. and Balasubramanyam, V., 1974. Storm surges in the Bay of Bengal. Q. J.R. Meteorol. Soc., 100: 437-449. Dube, S.K., Sinha, P.C. and Rao, A.D., 1982. The effect of coastal geometry on the location of peak surge. Mausam, 33: 445-450. Flierl, G.R. and Robinson, A.R., 1972. Deadly surges in the Bay of Bengal: dynamics and storm-fide tables. Nature, 239: 213-215. Henry, R.F. and Murty, T.S., 1982. Tides in the Bay of Bengal. In: G.A. Keramidas and C.A. Brebbia (Editors), Computational Methods and Experimental Measurements. SpringerVerlag, New York, pp. 541-550. Jelesnianski, C.P., 1965. A numerical calculation of storm tides induced by a tropical storm impinging on a continental shelf. Mon. Weather Rev., 93: 343-358. Johns, B. and Ali, A., 1980. The numerical modelling of storm surges in the Bay of Bengal. Q. J.R. Meteorol. Soc., 106: 1-18. Johns, B., Dube, S.K., Mohanty, U.C. and Sinha, P.C., 1981. Numerical simulation of the surge generated by the 1977 Andhra cyclone. Q. J.R. Meteorol. Soc., 107: 919-934. Johns, B., Dube, S.K., Sinha, P.C., Mohanty, U.C. and Rao, A.D., 1982. The simulation of a continuously deforming lateral boundary in problems involving the shallow water equations. Computers and Fluids, 10: 105-116. Murty, T.S. and Henry, R.F., 1983. Tides in the Bay of Bengal. J. Geophys. Res., 88: 6069-6076. Reid, R.O., Vastano, A.C., Whitaker, R.E. and Wanstrath, J.J., 1977. Experiments on storm surge simulation. In: Goldberg et al. (Editors), The Sea. Wiley-Interscience Publication, New York, London, Sydney, Toronto, 6: 145-168. Thacker, W.C., 1979. Irregular grid finite difference techniques for storm surge calculations for curving coast lines. In: J.C.J. Nihoul (Editor), Marine Forecasting. Elsevier Ocean Ser., pp. 261-283.