Proc. Indian Acad. Sci. (Engg. Sci.), Vol. 4, Pt. 3, September 1981, pp. 269-276. r Printed in India.

Storm surges in the Bay of Bengal

P K DAS Meteorological Office, New Delhi 110 003

MS received 14 February 1981

Abstract. In this paper, the scaling of the shallow water equations in terms of the speed of a cyclone moving over a basin, and another velocity related to basin charac- teristics is reported. The zero-order equations for which numerical solutions were obtained earlier are described. For a basin of uniform depth, an analytical solution has been obtained for the zero-order equations. The solution suggests that, in the initial stages, the divergence of the wind stress is more important than the curl of the wind stress. In subsequent stages, the curl becomes more important. It is also shown that the basin does not respond to a cyclone until a certain time has elapsed. The adjustment time depends on the storm speed, and is inversely proportional to the square-root of the basin depth.

Keywords. Surges; basin response; Bay of Bengal

1. Introduction

Storm surges represent an accumulation of water in shallow seas. This is brought about by the wind stress, and a reduction in atmospheric pressure due to a tropical cyclone. They are known to cause devastating floods in the coastal regions of India, Bangladesh, Burma and Thailand. Wherever there is danger of floods, it is useful to construct a design surge, which will be generated by a storm with specified charac- teristics. Mathematical models often provide useful insight.

This paper examines how a basin responds to a sudden impulse of the kind provided by a tropical cyclone.

2. Basic equations

The shallow water equations given below are used:

ut + (uux + vu~) - - f v = - - g ~ "-[- + ~----h '

-o-~ p, r t 7 / '

~, + [(~ + h) u], + [(~ + h) o], = O.

A list of symbols appears at the end of the paper.

(1)

(3)

269

270 P K Das

Suffixes have been used to denote partial derivatives. In deriving (1) to (3) quanti- ties such as ~ -- ha ___ 0 have been assumed. This assumption is generally valid save in exceptionally shallow waters. The basic equations reduce a three-dimen- sional problem into a two-dimensional one by using depth-averaged velocities. Although this does not provide information on the vertical structure of the surge, the equations are useful for examining the response of a basin.

For sealing the equations, we say

L = characteristic length (500 kin),

T = characteristic time (24 hr),

Z = 2 m ,

H = 10m,

V~ = average speed of storm propagation (6 ms-l),

C = Characteristic velocity of surge.

The vertical velocity at the free surface (W) is of the order of (Z/T) but, from the equation of continuity, we find

W]H ,~ C/L,

whence C N (L/H) (Z/T).

Thus, the magnitude of C is 1-2 ms -x, which is about a third of the storm speed (V~).

The speed of the storm may be used to define a Froude Number Fr. We assume

Fr = Vd(g H) 1/~. (4)

Considering average values we find Fr --_ 0.6.

It is convenient to make the equations non-dimensional. Let

f , ---- f T , ~, = g/Z, P . = po/pgH,

u , , v, = u, v/C, h , = h/H, AF. = AF/pu~,

x . , y . = x, y/L, t . = t iT AG, = AG/pu~,.

We have scaled the differential friction terms (AF[p) and (AG/p) by a characteris- tic friction velocity (u,). For strong winds, the magnitude of the friction velocity u, is approximately l0 -3 m z s -~. The non-dimensional variables are of order unity.

On dropping the asterisks, the non-dimensional equations are

(5)

Storm surges in the Bay o f Bengal 271

F 5 e ] lug, r1 so vt + E (uvx + vvy) + f u : 1 ~ + + - - F r --q I. C J, ~ ' ~ ] h ~ '

(6)

where

~, + [(,~ + h)u]x + [ (4 + h)v], = O,

, == Z / H : CT/L.

(7)

(8)

3. Zero-order equations

If we expand the dependent variables in (5), (6) and (7) in powers of ~, we may derive systems of increasingly higher order. The zero-order equations, when c < 1, are

f 5el /u, r Ae ut - - f v = - - 1 g + + Fr --'~ L C .Ix k-C---HI "-h '

(9)

1 5e] + [u:r ao (lO)

r, + (hU)x + (by), = 0. (11)

Numerical solutions of the zero-order equations, or their equivalents, have been obtained for the northern sectors of the Bay of Bengal (Das et al 1974). We assume an idealized storm, with a specified distribution of pressure and winds, which moves along a straight track with constant speed. This hits a given sector of the coast. By changing the storm characteristics, it is possible to relate the surge ampli- tude with the intensity and speed of the storm. An extension of this method for the Gulf of Thailand by Sinha (1981) is shown in figures 1 and 2. The interesting fact emerges that for the same storm characteristics the response of the Gulf of Thailand is different from the Bay of Bengal. Both basins have a narrow con- cave configuration on their northern sectors, but the bottom topography is different. In the northern part of the Bay of Bengal, there is a comparatively large area where the sea is extremely shallow. This feature is not observed in the Gulf of Thailand; consequently, the amplitude of the surge is smaller for a storm of the same intensity. The correct formulation of sea-bed friction and wind stress and boundary conditions on the open sea boundary are some of the difficulties encountered in numerical models. In general, quadratic friction laws are assumed, but it is not yet clear whe- ther the same friction law should be valid for both the storm surge and the astro- nomical tide. Tide-surge interactions are important, but difficult to estimate in shallow seas. The tidal cycle, often imposed as an initial state, cannot be determined with much accuracy due to lack of observations on the open sea. Despite these limi- tations, the zero-order equations provide a useful base for modelling a storm surge. Reliable observations of the storm surge are difficult to come by, but the available data suggest that models based on the zero order' equations provide fairly good

272 P K Das

IO0~

10

1 0 0

05~

O'N

5~

105" E Figure 1. Gulf of Thailand. Sea-bed topography (m). The storm track is shown by full line with arrow. Surge amplitudes were computed at four points along coast marked I, H, III and IV (after Sinha 1981)

estimates of the storm surge, except in very shallow waters where c is of the order of unity.

4. Basin response to winds

The zero-order equations could be used to estimate how quickly a basin will respond to the cyclonic winds of a tropical cyclone.

We may eliminate u, v from (5), (6) and (7) by computing the vorticity and diver- gence of the water. I f we assume a basin with constant depth (h) we find

[' ' I (")1 ~,,, + ~ - - Fr--- ~ V 2 ~, = ~rr2\-~l V 2 P, -t- V. -~ ,

02)

Storm surges in the Bay of Bengal 273

f " 11 I

I 1.2- /

/ /m

O . 4 -

o

-0.4,

--0,.8

16 20 24 28 32 3 6 40 46 time (hr).

Figure 2. Surge amplitudes at I, II, III and IV (after Sinha 1981)

where A~" ----- i A F ~ j AG

and i, j, k are unit vectors in cartesian coordinates.

As we can now see, the surge is generated by three forcing functions, namely, (i) the pressure tendency in the field of the cyclone, (ii) the rate of change of divergence of A ~- and

(iii) the curl of differential friction.

As sea-bed friction is a small fraction of the wind stress, the second and third forcing terms in (12) represent (i) the rate of change of divergence and (ii) the curl of the wind stress. We note the interesting fact that it is only the curl of the wind stress that has no time dependency.

Let us represent the Laplace transform of ~ by

O0

i = [ [ exp (-- st) tit, (13) 0

274 P K Das

with similar expressions for the transforms of P and Ar/p. We represent them

by P and AT/p. A distinction can be made between two situations. We have

(a) small values of time (t) when s--> oo, and (b) large time (t) when s -+ 0.

We find, after a little algebra, that for (a) or small time, the relevant equation for

is

where

\--h-/

=--h - h - \ C H I ", p ~

(14)

The subscript 0 denotes the initial value of [ at the beginning (t ---- 0).

For Co), or large values of t, we have

Vsv, Po, where 4,1 = V~ ~0 q-

(;)0] (u ,rlk v • + 4,z : \CHI t p '

4'3 = f ~ [ o q'- [~"

(15)

(16a)

(16b)

(16c)

A comparison between (14) and (15) suggests that for small values of t, that is, at the beginning, the surge is influenced by (i) the divergence of the wind stress, (ii) the pressure gradient weighted by the ratio (Vs/C) and (iii) the initial value of ~ weighted by Fr~/h. The curl of the wind stress does not come into play initially. But, after a certain length of time the divergence is no longer important, and the curl takes over as in (15). At longer times the gradients of the initial values of g and P appear to assert themselves, while in the beginning only go is important. These inferences are physically reasonable, because the initial response of a basin would be to drive away the water from the disturbance source. This is measured by the divergence of the wind stress. Subsequently, the rotational character of the wind will beoomo more important. This is the curl of the wind stress.

We may express the solution of (14) in-terms of a Green's ftmetion when we consider an infinite dometin. If

R = [(x - - xo) ~ + ( y - - yo)~] 1/~ (17)

Storm surges in the Bay of Bengal 275

is the distance between an observation point (x, y) and a source (x 0, Y0), then, for small R, we have

oO

.f J" I<o t rsR/ /hJ + (xo, :,o) dxo aye, - - 0 0

(18)

where Ko[Frsg/~v/-h] represents a modified Bessel function of the second kind. This is the Laplace transform of

[t 2 _ Fr 2 R21h]-l/3,

for t > FrR/~/-h (Morse & Feshbaeh 1953). A

As ~ is the Laplace transform of ~, we infer that the basin does not respond to the impulsive force of the cyclone until a time FrR/V'his reached. This time, as we can see, depends on the storm speed represented by the Froude Number (Fr) and is inversely proportional to a/h where h is the basin depth.

5. Summary and conclusions

The main conclusions of this study are the following:

(i) Sealing the shallow water equations in terms of the storm speed (Vs) and the characteristic surge velocity leads to a small parameter E which enables us to expand the dependent variables in powers of ~.

0i) The zero-order equations may be solved numerically for a basin with specified depth contours, coastal geometry and storm characteristics. The equations are sensitive to rapid changes in depth and coastal configuration.

(iii) An analytical solution for a basin of constant depth suggests that, in the initial stages, the divergence of the wind stress plays the dominant role. In the later stages, the curl of the wind stress becomes more important.

(iv) The rate of change of cyclone pressure is more important in the initial stages. (v) The basin does not respond to a cyclone until a certain time (FrR/V'h) has

elapsed. This time depends on the storm speed, and is inversely proportional to v'h where h is the basin depth.

List of symbols

f F~,F~

h

H proc.~2

Coriolis parameter (212 sin ~)

eastward component of the wind stress at the surface (FD and sea-bed friction (Fb)

northward component of surface stress (G~) and bottom stress (Gb)

depth of water

mean depth of water (10 m)

276 P K / ) a s

Pa atmospherio pressure

u, v depth-averaged velocities in rectangular cartesian co-ordinates (0 x y z), where the x and y axes point to the east and nor th respectively and z is the vertical axis

Z mean surge amplitude (2 m)

AF F, - - Fb

Aa G,-- a~

p density of water

amplitude of the surge

Figure 3 provides a physical representation of the above symbols.

l - - O

Z ~ - h

h M

Figure 3. The storm surge. Bottom thick line indicates sea-bed topography.

References

Das P K, Sinha M C & Balasubramanyam 1974 Q.J.R. Meteorol. Soc. 100 437 Morse P M & Feshbaeh H 1953 Methods of theoretical physics, Int. Student Ed., (New York:

McGraw Hill) p. 1360 Sinlm M C 1981 M ~ 3 2 4