Numerical experiments on tides and surges with DUCHESS model

Delft University of Technology Department of Civil Engineering Group of Fluid Mechanics

Numerical experiments on tides and surges with DUCHESS model.

PROJECT REPORT Delft University of Technology Department of Civil Engineering Group of Fluid Mechanics

Project title NORAD IND-013

Project description Transfer of ocean modelling capability to the National Institute of Oceanography of India

Customer Ship Research Institute of Norway

Marine Technology Centre Trondheim, Norway

represented by O.G. Houmb

Project leader dr. L.H. Holthuijsen

work carried out by dr. N. Bahulayan dr. N. Booij

Conclusion

Results of numerical experiments on tides _and storm surge in the North Sea, Bayof

Bengal and Arabian Sea with the Duchess model are presented. A detailed descrip-tion of the parametric type of cyclone model developed for Duchess, along with its documentation and source listing is also given in this report. The model simulated reasonably realistic values for tides, but it gave slightly higher values for surges. It needs proper calibration in the case of surge computation.

Contents

page

I INTRODUCTION ₁

11 THE DUCHESS MODEL 3

111 COMPUTING FACILITIES AT THE DELFT UNIVERSITY OF TECHNOLOGY

DELFT 9

IV BRIEF REVIEW OF TIDE AND STORM SURGE RESEARCH OF BAY OF

BENGAL AND ARABIAN SEA 10

V CYCLONE WIND MODELS FOR STORM SURGE COMPUTATION

A

Brief review of wind models

12 12 B Cyclone wind model for the computation of surges in

the Bay of Bengal and Arabian sea

C Implementation of cyclone wind model in the Duchess

15

17

VI VARIOUS CASES OF DUCHESS MODEL TESTS FOR TIDES AND SURGE 19 A Test on innundation procedure

B Simulation of tides in the North Sea with constant bottom and realistic bathymetry

19

19

c

Simulation of ~ component of tide in the Bay of Bengal

using realistic bathymetry 21

~

D Simulation of tide and surge in the Arabian Sea with

Duchess model using a constant bottom of 1000 meters 22

VII Summary and conclusions 24

Acknowledgement 25

References 26

Appendix 28

I INTRODUCTION

Numerical simu1ation of tides and storm surges has been an active area of research in many countries of the wor1d during the last two and a half decades. Considerab1e success has been achieved in this field of research by many advanced nations. This prob1em is becoming increasing1y important to coasta1 states as the coasta1 regions are now being deve10ped for industria1, residentia1 and recreationa1 uses. The design of permanent faci1ities at or near the shore1ines must inc1ude the consideration of maximum and minimum tides that may occur as we11 as wave action. The

insta11ation of tida1 power p1ants near coast1ines must a1so be based on the scientific prediction of tida1 heights of the region concerned. India has a vast coast1ine of more than 6000 kms 1ength and tida1 heights

varying from 1ess than a meter to more than 10 meters are observed a10ng its coast1ine. Tida1 heights of the order of 10 meters are norma11y observed near the gu1f of Cambay, Kutch and Coasts of Sundermanns of the Indian coast. The Government of India is now serious1y considering to estab1ish tida1 power p1ants near those regions. So the realistic prediction of tides is increasing1y becoming important to India.

The Bay of Bengal is considered to be the genesis area of many cyc10nes which norma11y strike the east and nothern coast1ines of India during the premonsoon and postmonsoon seasons, resu1ting in heavy 10ss to 1ife and property. Better prediction of the storm surges a10ng this coast can considerab1y reduce the 10ss to 1ife and property. Few hydrodynamica1 numerical mode1s of storm surges have been deve10ped for the Indian coasts, but with 1imited success. This is main1y due to insufficient

incorporation of the pecu1iar geophysica1 characteristics of Bay of Bengal in those mode1s. The geophysica1 aspects of Bay of Bengal are quite unique for storm surge computation in the sense that it is a deep and a1most semicircu1ar basin with a fair1y narrow continenta1 she1f a10ng the east coast of India. The width of the continenta1 she1f is 1ess than 20 kms in certain regions of this coast. So storm surges which are usua11y unnoticed in the open ocean are sudden1y amp1ified on approaching the coast1ine. Thus, it is essentia1 to deve10p/adapt a numerical model of storm surges

that wi11 take into consideration the actua1 bathymetry of the region. The model shou1d a1so be capab1e of predicting the surges for areas with a high1y irregu1ar coast1ine such as del tas or 10w 1ying coasta1 p1ains with irregu1ar topography.

The main objectives of this project are to adapt a numerical model for tide and storm surges for Indian coasts which would consider the peculiar geophysical characteristics of the region and also to modify this model to include a cyclone wind and pressure model. Delft University of Technology has a non-linear model for tides and storm surges, cal led DUCHESS, which is being adapted for the simulation of tides and surges along the Indian coasts in this report. Chapter 11 of this report mainly deals with basic characteristics of the Duchess model, namely model equations and numerical schemes. A brief de script ion of the computing facilities at Delft

University of Technology is given in Chapter 111. Some numerical models have been developed for the Indian coasts and a brief review of the same is given in Chapter IV. As mentioned earlier, one of the objectives of this program is to develop a cyclone model for Duchess. So a brief review of various cyclone models and the cyclone model used for Duchess are given in Chapter V. The complete program documentation along with listing and test output of the cyclone model is also given as an appendix at the end. Chapter VI deals with the result of the computation of tides and surges for the North Sea, Arabian Sea and Bay of Bengal.

11 THE DUCHESS MODEL

(i) Important facilities of Duchess model

The Duchess model is mainly intended to perform two-dimensional computation of tides and surges in estuaries and coastal waters. It is based on two-dimensional shallow water equatio~s which are solved

numerically by fini te difference method. The computational procedure is according to Alternating Direction Implicit (ADI) method.

The following are the important facilities of Duchess model.

(a) Tides and surges can be computed by means of steady and unsteady type of boundary conditions in either water level or one of the flux

components. The model can handle a very irregular type of bottom topography. Dams in the computational region can be modelled by means of internal boundary conditions.

(b) The model allows nesting of coarse and finer grids in the computatio-nal domain. The boundary data for the finer grid model can be obtained from the results of a coarse grid model.

(c) Surges can be simulated by means of steady and unsteady wind forcing. The forcing can be given through wind stress and/or air pressure. (d) The program also allows certain parts of the computational domain to

become dry or to get innundated during numerical integration. (e) The input to Duchess is given by means of user-friendly commands. (f) The Duchess model has several output facilities. Output can be

obtained in numerical form to a printer and also to secondary storage units to use as input to other models. Output can also be obtained in the form of vector plots of flux and velocity and isolines of water depth and water elevations.

(ii) Model equations

The Duchess model is based on two-dimensional longwave equations which are integrated in the vertical direction. Thus, the equations appearing in the model are functions of horizontal co-ordinates x and y and of the time t. The unknowns appearing in the equations are H, the water level with respect to a chosen datum, Qx and Qy, the x and y-components of velocity integrated over depth. All the unknowns are taken positive upwards from the chosen datum. There are three equations, one of them is

the continuity equation which follows from the conservation of masso

(1)

The x- and y-components of vertically integrated momentum equations can be written as follows

a a 2 a a a a a

at(Qx) + ax(QX/D) + ay(QXoQY/D) - ax(DoE ax(Qx/D)) - a/DoE ä;(QX/D))

+

gD

l.

(H

+

P)

+

F IQIQx0 D-2 - C Qy

+

W = 0 ax r 0 x (2 ) d d ay (D oEay (Qy/D))

+

gD ..ê..(H

+

P)

+

F IQIQyoD-2

+

C Qx

+

W = 0 ay r 0 y (3)

The various terms appearing in equations (1) - (3) are the following: - local acceleration terms

- advective acceleration terms - viscosity terms

- slope and pressure terms - bottom friction terms

- coriolis force and wind shear componentso

The variables in equations (1) - (3) have the following meanings:

=

gravitational acceleration water depth (H - Z)

=

bottom level

=

eddy viscosity coefficient

=

surface pressure divided by gravitation and water density

=

bottom friction coefficient

=

j

Q~

+

Q~ g D Z E p F r IQI C 0 W and Wy x

=

coriolis parameter

(lii) Numerical Scheme and Discretisation

The above mentioned equations (1) - (3) are solved by means of an Alternating Direction Implicit method. Current vector and water levels are calculated at alternate points. In the computational procedures, new

values for Qx, Qy and Hare calculated at every half time step. In the first half time step computation along the x-direction takes place and in the second half time step computation along the y-direction. In the

computation along the x-direction, the derivatives with respect to x are treated implicltly and the derivatives with respect to y explicitly and vice versa. The water levels are advanced in time using the continuity equation, the currents using the equations of conditions. The partial differential equations are approximated by means of a numerical scheme which is central in space and approxlmately central in time.

The finite difference approximations to partial differential

equations (1) - (3) are given below. Only computation in the x-direction is described since computation in the y-direction is ldentical, apart from the swapping of the variables Qx and Qy. The superscript - indicates the known value at time T, the superscript

+

indicates the as yet unknown value at a time half step ahead (T

+

ST/2).

In the first half time step (implicit in x-direction), the continuity equation (1) is approximated by the following expressions

+

-

+

+

(Hi .- Hi

.)/(ST/2)

+

R(QXi .- QXi-l

.)/SX

,J

,J

,J

,J

+

(1 - R) (Qxi,j - Q~i_l,j)/SX

(4)

The equation for the computation in the second half time step is found by swapping Qx and Qy and by substitutlng (l-l,j) by (l,j-l), etc.

R

=

28, where 8 is a coefficient lylng between 0.5 and 1 whlch ls used to control the amount of numerical damplng.

The equation of motion in x-directlon (implicit ln x-direction) is approximated by:

(Q~. . - Qx. .) / (ST/2)

+

1.,J 1.,J { 2R' (Qt .

+

Qi. I .)

+

(1 - 2R' )(Qx ..

+

Qx. I

')J

1.,J 1.+ ,] 1.,J 1.+ ,J

(QXi,j

+ QXi+

1

,j)

.D'tl . • X 1. , J (Qx .. 1.,] + Qx.-1.+ ,JI •){,:+2 R (Qx.1.,J. + Qx.:+1.+ ,JI • )

+

(1-2R')

(Qx . . + Qx.

1.,J 1.+ ,]1 .)} 2D ••• Sx 1.,]

+

(Qx ..

+

Qx .. I) (QY..

+

QY1.·+I,J·) 1.,J 1.,J+ 1.,J

(D..

+

D

.

I . +

D ..

1

+

D. 1 .

I)Sy 1.,J 1.+,J 1.,J+ 1.+ ,J+

+

(D.

.

+

D. 1 .

+

D ..

I +

D. 1 .

I)Sy 1.,J 1.+,J 1.,J- 1.+ ,J-+

D. 2 .)/2

1.+ ,] :+ Qx._1.,J. } 2E. .' D. • 1.,] 1.,] Sx2 :+

{

QX.

I . 1.+ ,] (D. 1 • 1.+ ,J

(D..

1.,J +

D. 1 .)/2

1.+,] 2E .• D. 1 .

+

1. 1.-,] Sx2

{

Qii

(D. • 1.,J + D.1.+ ,JI .) /2 + QX. I . 1.- ,J (D.• + 1.,J

D. I .)/2}

1.- ,] g(D .•

+

D. I .)

+

---=~--~-.

1.,J 1.+,J (_{P. I . - P .. + R H.}( + _{I . - H ..}+ ) 2Sx 1.+,J 1.,J 1.+,J 1.,J

FrIQI(2Qi

J' -

Qi .)

+

"J 2 (D. . + D. 1 .) /4 1.,J 1.+,]

+

W

=

0 Xi,j (5)

In the above equation (5), the term R' performs a similar function as

of time T -

ST/2. This is necessary in order to maintain stability; using

the values of time T causes unstable behaviour. In the numerical

experi-ments R

=

1

is used for the computation.

The computation in y-direction (implicit in x-direction) is

approxi-mated by:

(QYi,j -

QY

i

,j)/ST/2)

2(Qx

i .

+

+

,J

QXi ·+l)·R'(QYi

,J

,J

.

+ QYi+l .)+(l-R')(QYi .+ QYi+

,J

,J

1 .)

,J

(D..

+ D ..

1 + D·I·

+ D·I·

1)·Sx

~,J

~,J+

~+,J

~+ ,J+

2.(Qx·

·

1 . + QX·

·

1 . 1)·R'(QY

i

1 .+Qy.. )+(1-R')(QY· 1 . + QY·

.

)

_ 1-

,J

~- ,J+

- ,J

~,J

~- ,J

~,J

(D·I·

+ Di 1 . 1 + D. . + D. . 1).Sx

~- ,J

+ ,J+

~,J

~,J+

(QY·

.

+ QY· . }).(Qy . . + QY· . 1)

+

~,J

~,J+

~,J

~,J+

2.D. .+1 • Sx

~,J

(cty . . + QY

·

. }).(Qy . . + QY· . 1)

_

~,J

~,J-

~,J

~,J-

.

2.D. . • Sx

~,J

E - --2 (D.

_~,J

.

+

D.

_~,J+

. 1

+

D·I'

+

D.+ 1 .+ ] ) r+

,J

~

,J

2.Sx

D. 1 .+D. ]

.

}

~+,J

a+

,J+

Qy ..

~,J )

D .. +D ..

I

~,J

~,J+

+

E ₂

_{(Di,j + Di,j+ I + Di-) ,j + D i-I,j+ 1)•}

2.Sx

- QYi+l,j

)

D. I .+D. }

·

+1

~-,J ~-, J

D .. +D ..

l

~,J

~,J+

(D. .

+

D. . 1)

~,J

~,J+

2.Sx

+

g F •

SQRT

(QX . .

Qx .

.

+

QY·

.

Qy . .)

Qy

.

.

+

r

~,J

~,J

~,J~,J

~,J

2 (D . . + D. •+1)

14

~,J. ~,J

+

C •

Qx

+

Wy. . =

0

~,J

(6)

Analogous equations are used for the computation implicit in y-direction. The Qx and Qy values, as weIl as x and y-coordinates are interchanged. In the manner described above, a set of linear equations are

+

+

formed with Hand Q values as unknowns. This system is solved efficiently by means of Thomas algorithm and new values for Hand Q

result. This procedure is followed for every line in x-direction. Then the time is increased by a half time step and the procedure is applied in y-direction.

111 COMPUTING FACILITIES AT THE DELFT UNIVERSITY OF TECHNOLOGY DELFT

The Delft University of Technology has a fourth generation mainframe computer, named AMDAHL 470V/7-B, with a user available memory of

5 M Bytes. It also has a virtual memory unit of 16 M Bytes and as such programs that require memory storage up to 16 M Bytes can be run on the mainframe. The mainframe is connected to a number of minicomputers that are fixed at the various departments of the University.

The mainframe computer has compilers in Fortran IV and 77. In addition to minicomputers, a large number of terminals and lineprinters are

connected directly or via cluster controllers to the mainframe. Outputs of computations can be obtained either in the form of hardcopies (print

output) or plots. It can be routed to local line printers that are directly attached with the mainframe or remote line printers which are fixed at the various departments of the University. Plotting is done using a remote TECTRONIX terminal. The mainframe has 48 IBM 3350 disc units, 6 tape units with densities of 800 BPI, 1600 BPI and 6250 BPI. lts execution speed is of the order of 3.8 mips (million instruction per second).

The editing/job scheduling system is cal led VSPC (Virtual Storage Personal Computing). Scheduling and fetching of jobs are done by means of user friendly general and local VSPC commands. Editing of jobs can be done either in input or view mode, but no corrections in the program can be made when the system is in input mode. Another peculiarity of the system is that it accepts jobs only in the batch mode.

All the numerical experiments on tides and storm surges with the Duchess model were carried out by making use of the facilities available at the computer centre of the Delft University of Technology Delft and at its Department of Civil Engineering.

IV BRIEF REVIEW OF TIDE AND STORM SURGE RESEARCH OF BAY OF BENGAL AND ARABIAN SEA

Hydrodynamical Numerical modelling of tides and storm surges in

estuaries and coastal waters has been an active area of research for quite some time and considerabie amount of Iiterature is available in these fields. But very little information on tides is available for the Bay of Bengal and Arabian Sea except some of the recent publications on tides by E.W. Schwiderski (1979), T.S. Murty et al. (1983). It is to be pointed out that none of the above mentioned módels were calibrated with actual tidal observation at closely speed intervals. Accurate knowledge of tides is very important, as the worst disasters occur when large surges coincide with high tide. So as a first step towards predicting the combined surge and tide effects, it is essential to model the tides first in the absence of surge effects.

Schwiderski (1979) used a unique hydrodynamical interpolation technique to compute partial global ocean tides in great detail. This technique has been applied to construct the leading semi-diurnal principal lunar (M2) ocean tides in the open oceans. The resulting tidal amplitudes and phases are tabulated on a 1°x1° grid system covering the whole oceanic globe. Such global tide models are not capable of simulating tides in the estuaries and coastal regions.

Murty and Henry (1982, 1983) simulated the amplitudes and phases of tidal constituents

Mz,

S2' Kl and Ol of the Bay of Bengal with the aid of numerical models based on vertically integrated equations of motions and continuity. The simulation results were compared with tide gauge records at 112 locations in the Bay of Bengal. Generally good agreement was found between numerical model results and observations.

The coastlines of India are vulnerable to storm surges. Maximum storm surges are observed along the east coast of India during the premonsoon and postmonsoon seasons. In 1970, a severe cyclone from the Bay of Bengal that passed over Bangladesh generated a surge exceeding

6

meters. Das (1972, 1974) was the first who attempted to simulate the surges in the Bay of Bengal by means of numerical models. He used a linear shallow water model to evaluate the surge. The surge amplitudes were computed for different values of storm intensity and speed. It has been found from numerical experiments that the superposition of the surge on tide leads to an overestimate of sea level at the time of landfall.

Ghosh (1977) used Jelesnianski's SPLASH model (1972) to prepare nomograms for estimating the peak surges generated by tropical cyclones impinging on the east coast of India. The nomograms accomodate fixed

values for pressure drop, radius of maximum wind, vector motions of storms and offshore bathymetry. The study does not deal with curved coastline or with storms that move alongshore.

Johns and Ali (1960) developed a non-linear storm surge model for the Bay of Bengal. The model allows determination of interactive effect

between surge and astronomical tide. Model incorporated the river system adjoining the coasts and showed that surge may penetrate deep inland leading to flooding hazard in the inland waterways of Bangladesh. Modelling experiments also showed that the surge response depends

critically on the track and diameters of the forcing cyclone. The effect of barometric forcing has not been included in the model and only wind stress forcing was used.

Johns et al. (1981) used their different numerical models to simulate the surge generated by the 1977 Andhra cyclone. One of the models was based on a cartisian co-ordinate system whereas the other two were based on curvi-linear co-ordinate systems. These models were non-linear and could accomodate flooding and drying. However, it was found that the models were not suited for areas with highly irregular coastlines such as de1tas or lowlying coastal plains with irregular topography. Using

available data on Andhra cyclone as input to numerical modeis, Johns et al. (1981) compared the predicted rise in sea surface elevation

obtained from each of those with estimates based on reports received from Andrapredesh af ter the disastrous flooding in that areas. Each of the

three models produced a quantitatively similar surge response. It is to be mentioned that none of these model results were compared with actual tide gauge records. Johns et al. (1982) developed a three-dimensional numerical model for the simulation of storm surge generated by tropical cyclones off

the east coast of India. Numerical experiments were conducted using wind stress forcing data representative of 1977 Andhra cyclone and the results were compared with earlier simulations using depth averaged modeis. It has been found that even with more refined treatment of frictional processes there was no substantial difference between the simulations performed with two and three-dimensional modeis.

V CYCLONE WIND MODELS FOR STORM SURGE COMPUTATION

A Brief review of wind models

Storm surge models use windstress and pressure as input. Devastating storm surges are normally caused by strong winds with an onshore

component. When atmospheric disturbances such as storms or cyclones move over oceanic areas, energy is transferred to the sea through tangential and normal stresses which are generated by pressure gradients. This input of energy is partly lost by tangential stresses at the seabottom. Most of the energy supply is initially kinetic and several processes take place before surges (potential energy) are generated and maintained. So the computation of wind field in a cyclone or storm to a certain degree of accuracy is an essentail prerequisite for the realistic simulation of storm surges along the coasts.

Various methods are used for the computation of windfield in cyclones and storms. These methods can be briefly summarized as follows:

Geostrophic method

The geostropic wind is an approximation to the actual wind at the sea surface level by which the direction and width of isobaric channels are converted into wind velocity. The approximations implies the following simplifications:

(i) the curvature of the isobars are neglected (ii) the motion is assumed to be free of friction

(iii) the windspeed changes so slowly that acceleration along the path may be neglected.

The above mentioned three assumptions imply that the horizontal 1 dO

pressure gradient force -~ is balanced by the coriolis force V2 sin, so p on

that

V= 1

2.E.

p2r2sin<l>dn (1)

where

V - geostrophic wind speed

~ - pressure gradient normal to isobars dn

p - air density <I> - latitude

The balance between the coriolis force and horizontal pressure gradient as given by equation (1) is applicable only in large scale currents above the earth's friction layer. In fact, surface friction considerably reduces the wind speed ne ar the sea surface level. Surface wind can be estimated by reducing the geostrophic wind by a fixed percent. In a cyclone isobars are nearly circular in shape and hence the

geo-strophic method cannot be employed for the computation of wind field.

Gradient wind method

When an air parcel is travelling in a curved path over the earth's surface, it experiences a continual centripetal acceleration towards the centre of curvature of its path. So a balance is maintained between the centripetal acceleration and sum of the pressure gradient and horizontal deflecting force of earth's rotation. This wind is cal led gradient wind. It can be expressed as follows:

v

2

- JE.

-

- ..!.. ~

+

2rN sin<j> r p dn gr (2) where V gr r dp dn

n

<j>

- gradient wind speed

- radius of curvature of the air parcel - pressure gradient normal to isobars - angular velocity of earth's rotation - latitude of the place

The solution of equation (2) would give gradient wind speed in a cyclonic field which depends on the radius of curvature r, of the air parcel, latitude <j>,of the place, the air density pand horizontal pressure gradient ~~

Cyclostrophic wind method

In an intense tropical cyclone, the centripetal acceleration is as large as the horizontal pressure gradient force and the horizontal

deflecting force of earth's rotation is neglected. Such a wind is called cyclostrophic wind. The magnitude of the cyclostrophic wind is obtained by equating the centripetal acceleration and pressure gradient force.

i.e. V2 c - - =-r J dp

P

dn or

v

=!E§i

_..f

_p

_an

(3) where

v

c r

- cyclostrophic wind speed

- radius of curvature of air parcel - air density

- pressure gradient normal to isobars

p dp

dn

The gradient wind and cyclostrophic wind equations give bet ter values of wind speed within the radius of maximum wind. At larger distances from the radius of maximum wind, the angle of vector winds across the circular isobars needs consideration and various types of empirical relationship are available for computing wind field in a cyclone.

In some cyclone models, the wind profile is specified (Jelesnianski, 1972) by the following relationship:

where

Ver)

=

VCR)

x

2Rr

R2

+ r

2

(4)

r

- radius of maximum winds

- the radial distance from the storm centre - maximum wind speed

R

VCR)

This profile increases the wind speed from r=0 to a maximum at r=R, but thereafter it again decreases it back to zero. Using equation (4) and equations of motions for a stationary storm (Myers and Malkin, 1961), it is possible to determine the angle of inflow and its variation with r. The angle varies from zero at storm centre to about 30° at a radial distance of 3R and is thereafter practically constant.

A number of other numerical models (Isozaki, 1970; Das et al. 1974) specify pressure field by one of the following relations:

per) = _{peet)- ~} a (5) (*)2]

i

+ per)

=

1010.0 - a (6) +·(!.)2 where R

per) and peet) represent sea level pressure at 'r' and at the cyclone peri-phery, 'R' is the radius of maximum winds and 'a' is the pressure drop.

The second expression proves a more rapid decrease in pressure with rand also uses a constant value (1010) for the pressure at the periphery of the cyclone. The cyclostrophic wind for (6) can be written as follows:

(7)

where ~

= ~

_R and V_m is the maximum wind speed at R. The maximum wind and pressure drop may be related by

V

=

c x (a)~

m (8)

where c is a numerical constant.

B Cyclone wind model for the computation of surges in the Bay of Bengal and Arabian sea

A parametric type of relationship, suggested by Jelesnianki (1965), is used in conjunction with the Duchess model for the computation of surges in the Bay of Bengal and Arabian sea. These formulas are extensively tested for the simulation of surges along the United States coasts. The main cyclone parameters that are required for the computation of windfield in a cyclone are the following:

(a) position of the eye of the cyclone in latitude and longitude (b) radius of maximum wind

(c) central pressure of the cyclone

(d) pressure at the outer periphery of the cyclone (e) maximum wind speed in the cyclone

If V_max is the maximum wind speed in the cyclone, R, the radius of maximum wind, the wind speeds at a radial distance r from the centre of the cyclone are given by the following relationships

V • V (:..)3/2 _{for r}

<

R (9 )

r max R

V_r '"' V (!) _{for r >R (10)}

max r

The maximum wind speed in the cyclone can be correlated with the pressure drop (difference between peripheral and central pressure at mean

sea level). For the Indian seas, Mishsa and Gupta (1976) related the pressure drop dp with the maximum wind speed in the cyclone by the following relationship.

v

=

14.2

$p

(ll)

max

In the above equation (11), V is in knots and p is in millibars. max

For the computation of surface wind stress, a quadratic friction law is used. The wind stress at the sea surface is nearly proportioned to the square of the wind speed.

i.e. T ,. P k air

Iv Iv

p r r water (12) where

k is the stress constant or T ...

Klv [v

r r (13) where . Pair 3 k ...k--- = 2.8x 10-Pwater

The components of surface wind stress Tand T at any point in the

x

y

cyclone field can be found out as follows. Let us consider a point p(x,y) on an isobar which is at a distance r from the eye of the cyclone, say

(xo' Yo) as shown in figure 1.

-T

From figure 1 we have x

₌

reos

e

y

=

rsin

e

r2

=

(x - x )2

+

(y _ y )2 0 0 We also have x

=

x

+

reose o y Yo

+

rsin e x - x c os f = 0 r (14) y - Yo sine = ---r (15)

From figure 2, the eomponents of surface stress 1 and 1 ean be

x y found. 1

= -

tsin e x (16) 1 ...1 cos e y (17)

Substituting the values of sine and eose from (14) and (15) into equations (16) and (17), we get the following relationship for the wind stress eomponents 1 x

=

-(y - y ) . (y - y )

T __

_;;o_

=

2.8 x

lo-3lv

[v

0 r r r r (18) (x - x ) (x - x ) 1 - 1 0_

=

2.8 x

lo-3lv

lv

0 y r r r r (19)

In the Duehess model, wind stress formulas (18) and (19) are used for the eomputation of surges.

The Jelesnianski (1965) paper also eontains a nearly-eonsistent pressure field model. It is not yet implemented in the cyc lone model.

C Implementation of eyelone wind model

In order to simulate the surges, it is neeessary to eompute the wind stress eomponents at the model grid points. Initially the eyelone

parameters sueh as time, eye of the eyelone in latitude and longitude (in degrees) and pressure drop (in millibars) are given as input to the

cyclone model. The radius of maximum wind in this cyclone model is computed by means of a linear relationship with the maximum wind speed

(Cray, 1978). The program linearly interpolates all the cyclone parameters such as time, position of the eye of the cyclone in terms of distances from output points, maximum wind speed in the cyclone and radius of maximum wind. The complete program, documentation and test output of the cyclone model are given as an appendix in this report.

One advantage of this model is that it considers the actual track of the cyclone during its complete life span and computes the wind stress fields at fixed time intervals and at grid points. Thus, the surge model is driven by an unsteady type of wind forcing which is normally not considered in majority of existing storm surge modeIs.

VI VARIOUS CASES OF DUCHESS MODEL TESTS FOR TIDES AND SURGE

A Test on inundation procedure

The inundation procedure is one of the test problems used to verify the correctness of Duchess program. The boundaries of the model area are shown in figure 1. There is a flat floor of 2 meters depth from the datum and a shoal is introduced into the model domain. The shoal falls dry partly during the numerical integration of the model. There is also a dam inside the model area on which normal flux boundary conditions are

applied. Harmonic type of boundary condition is applied at the open boundary for water level. The circulation pattern, developed af ter 16000 seconds of numerical integration, is shown in figure 1. It has been found that an eddy type of circulation is developed behind the dam which is normally expected in this type of situations. The simulated flow pattern, as shown in figure 1 clearly reveals that normal flux and water level boundary conditions are properly applied in the Duchess model.

B

Simulation of tides in the North Sea with constant bottom and realistic bathymetry

Oceanographically, North Sea is considered to be one of the weIl explored seas of the world. Activities associated with the product ion and exploration of oil have led to increased interest in the hydrography of the North Sea. Various types of numerical models of tides, currents and other physical processes of the North Sea were developed and calibrated by the countries adjoining it. It is to be mentioned that North Sea is one of the few seas of the world for which water levels and currents are

predicted by numerical models for routine operational purposes. There is also a large number of observational networks in various parts of North Sea to test the validity of the numerical models being developed. In view of these facts, we decided to test the Duchess model for the North Sea area first so that model results could be compared with observations and also with the results of other numerical modeis.

Tides in the North Sea were simulated by using a constant depth of 50 meters from the datum for the whole domain and also by using realistic bathymetry. The model area extends from 3°W to 12.5°E in the east-west direction and 59°N to 49°N in the north-south direction. There are 13 and 14 points in the x and y direction respectively with a grid size of

bathymetry used for testing the model is shown in figure 3. A periodic type of boundary condition for water level, given by fourier coefficient, was applied at the northern open boundary and also at the south-western open boundary near the English channel. The model was integrated for a maximum period of 3 tidal cycles at a time step of 4000 seconds. The vector plots of depth averaged veloeities at various times of numerical integration of the model are shown in figures 4, 5, 6, 7 and 8. The circulation in the North Sea is very complex due to the fact that the northern part of it joins with the Norwegian Sea and the south-western part joins with the English channel. There is also fresh water intrusion at the eastern part. So waves are penetrating into the North Sea from 3 sides. As shown in figure 4 astrong current is observed in the

sou th-western open boundary near the English channel where current speed of the order of 0.5 mis is observed. The circulation is weIl developed af ter the model is integrated for one tidal cycle which is clear from figure 5. The currents near the English channel are reversed af ter the model is integrated for 44000 seconds. The model has been found to be very stabie even af ter 3 tidal cycles of integration and the changing

circulation pattern af ter each tidal cycle is well noted in the figures.

The M2 tidal component variation at the northern boundary point A (refer figure 2) of the North Sea in relation to time is given in table I. The maximum elevation from the mean sea level for

Mz

tide is found to vary between 0.75 meters and 0.72 meters.

Numerical experiments were performed for the computation of

Mz

component of tide in the North Sea by using realistic bathymetry of the region. Depths were digitised for each of the grid points from figure 3. The model was driven, as in the previous experiments, by the harmonie boundary conditions applied at the northern and south-western part of the model domain. The model was integrated for a minimum period of 3 tidal

cycles with a time step of 4000 seconds. Results of computations at various time intervals are presented in figures 9-16. The isolines are lines of equal sea level elevation. The flux (m2/s) and velocity vectors are also shown in the form of vector plots.

Figure 9 shows the variation of M2 tidal component at the point A (refer figure 2) in relation to time. The maximum elevation is found to be approximately 78 cm at this point. This value is fairly in agreement with M2 tidal values available in the literature.

Table I M2 tide variation at the northern open boundary point A (refer figure 2) of the North Sea in relation to time.

Time in Water elevation in

seconds _{cm for ~ tide}

4000 -33 12000 -74 20000 -31 28000 +47 36000 +72 44000 +15 52000 -59 60000 -66 68000 +2 76000 +68 84000 +56 92000 -19 100000 -73 108000 -44 116000 +35

It has been found that there is a gradual decrease in the sea

elevation from west to east at the northern open boundary as shown in all the figures. Maximum elevation upto 1.62 meters is found at the

south-western open sea boundary. The circulation as in the previous

experiments is weIl developed af ter the model is integrated for one tidal cycle. Velocities of the order of 0.4 mIs to 0.5 mIs are observed near the English channel.

C Simulation of ~ component of tide in the Bay of Bengal using realistic bathymetry

The M2 component of tide in the Bay of Bengal was simulated by using the actual bathymetry of the region with the Duchess model. The model area extends from lOoN to 22°N in the north-south direction and 500E to 1000E in the east-west direction. There are 41 and 25 grid points respectively in the x and y direction with a grid size of 55 kilometers. The bathymetry of the region is digitised for each grid point from bathymetric chart and a computer plot of the same is shown in figure 16b. lt is found from

figure 16b that the bottom contours vary very irregularly in the reg ion with few meters near the coastline to 3500 meters in the central part of the Bay. The southern boundary of the domain is open where a harmonic periodic boundary condition for water level was given. The fourier

coefficients for the boundary condition were computed by making use of the publication on global ocean tides by E.W. Schwiderski, (1979).

The model was integrated for a period of 3 tidal cycles with a time step of 400 seconds and the results of ~ tidal computation for the point B (refer figure 16b) are shown in figure 17. Maximum sea elevation of 34 cm was found for this point for M2 component. This value is very much in agreement with the tidal elevation (32 cm amplitude) reproduced by E.W. Schwiderski (1979). M2 tidal values for a number of points in the computa-tional area were compared with existing literature on tides and it has been found that the Duchess model simulated fairly accurately the M2 com-ponent. It has also been found that this model could be used successfully for simulating tides in deeper and very irregular depth regions.

D Simulation of tide and surge in the Arabian Sea with Duchess model using a constant bottom of 1000 meters

As part of Duchess model tests on tides and surges, numerical

experiments were performed for tides and surges in the Arabian Sea with a constant bottom of 1000 meters depth. The model area extends from 110N to

26°N in the north-south direct ion and 56°E to 76°E in the east-west direction. There are 41 and 31 points respectively in the x and y directions with a grid size of 55 kilometers. There are three open

boundaries in the model domain, one in the south, one in the west and one open boundary point near the Persian gulf region. These open boundaries are marked in the figures 18 - 27.

As in previous experiments, a periodic boundary condition for water level was given at the open boundaries. The fourier coefficients for these open boundaries were computed by making use of the publications on global ocean tides by E.W. Schwiderski (1979).

The model was integrated for a period of two and a half tidal cycle at a time step of 2000 seconds. The results of computation for ~ components of tide along with vector plots of flux, isolines of water elevation and depth averaged velocities in relation to time are presented in figures 18 - 23. The sea level elevation at a point marked 'Cl in figure 18, off the

west coast of India, is presented in figure 24. As seen from these figures, a weIl developed circulation is formed af ter a few hours of numerical integration of the model. Tide waves penetrate to the central part of the Arabian Sea from the western and southern boundaries of the domain.

M2 component of tide at the point marked 'Cl (figure 18) simulated by the Duchess model, was compared with publication on tides by

E.W. Schwiderski (1979). Maximum tidal elevation of 31 cm was simulated by Duchess and this value is fairly in agreement with the above publication

(34 cm amplitude).

Numerical experiments were performed for surge in the Arabian Sea. The windstress values, representative of Daman cyclone, were used as input to

the Duchess. It has been found from figures 25, 26 and 27 that a cyclonic type of circulation is formed at the central part of the Arabian Sea. Too high surge values were observed af ter a few hours of numerical

integration. 80 the model has to be properly calibrated to compute the surges.

VII Summary and conclusions

Numerical experiments were performed with the Duchess model for the computation of M2 component of tide in the North Sea, Bay of Bengal and Arabian Sea. The model could successfully simulate the tides in these regions with a reasonable degree of accuracy. It is to be pointed out that the model needs proper calibration for the realistic simulation of surges.

Duchess model has also been found to be an effective numerical model for simulating the water level elevation in very deeper regions and in irregular topography.

Acknowledgements

The authors wish to thank prof. Battjes, chairman Fluid Mechanics group, Delft University of Technology, for having provided all the facilities during the implementation of this project NORAD-GOA/IND-013. The authors also wish to thank Dr. L.H. Holthuijsen, project leader, who has supervised this investigation from the very beginning. Helps rendered by Simon de Boer, system analysist, during the various stages of numerical experiments on tides and surges, are a1so greately acknow1edged.

This work was carried out as part of the NIO-NORAD collaboration

program on coasta1 zone management. In this connection, one of the authors (N. Bahulayan) wou1d 1ike to thank Dr. V.V.R. Varadachari, Director

Nationa1 Institute of Oceanography, Goa and Dr. B.U. Nayak, Head, Ocean Engineering Centre, NIO, Goa, for having deputed him to the Netherlands and Norway to imp1ement the Duchess model on tides and surges. The financial support of NORAD in imp1ementing this project is greate1y acknow1edged. He a1so thanks prof. O.G. Houmb, project leader from NORAD side, for the initiative and leadership in this project.

Dr. Gerritsen of the Delft Hydrau1ics Lab. is thanked for his help in obtaining and using the Schwiderski data; Mr. Roos of the Netherlands Public Works Dept. is thanked for making avai1ab1e the data used for the North Sea model.

References

1. Bryan Pearse and Vijay Panchang, 1983, 'Estimation of design wave heights in the gulf of Maine', University of Maine/University of NewHampshire, Sea grant college program, pp. 112.

2. B. Johns and M. Anwar Ali, 1980, 'The numerical modelling of storm surges in the Bay of Bengal', Quart. J. R. Met. Society (1980), 156, pp. 1-18.

3. B. Johns et al., 1981 'Numerical Simulation of surge generated by the 1977 Andhra cyclone', Quart. J. R. Met. Society (1981), pp. 919-934. 4. B. Johns et al., 1982, 'Simulation of storm surge using a

three-dimensional numerical model: an application to the Andhra cyclone', Quart. J. R. Met. Society, 1982.

5. Das, P.K., 1972, 'Prediction model for storm sturges in the Bay of Bengal', Nature, 239, pp. 213-215.

6. Das, P.K. et al., 1974, 'Storm surge in the Bay of Bengal', Quart. J. R. Met. Society (1974), 100, pp. 437-449.

7. Ghosh, S.K., 1977, 'Prediction of storm surges on the east coast of India', Indian J. Met. Hydro and Geophysics (1977), 282, pp. 157-164.

8. Henry, R.F. and T.S. Murty (1982), 'Tides in the Bay of Bengal: Computational methods and experimental measurements', Proc. J. Inter. Conference Washington Dc, July 1982, Edited by G.A. Keramidas, CABrebbia, Springer Berlin.

9. Flierl, C. and Robinson, A.R., 1972, 'Deadly surges in the Bay of Bengal', Nature, London, 239, pp. 213-214.

10. Isozaki, I., 1970, 'An investigation on the variation of sealevel due to meteorological disturbances on the coast of the Japanese

Islands', VI. Pap. Met. Geophys., Tokyo, XXI, 3, pp. 291-321.

11. Jelesnianski, 1965, 'A numerical calculation of storm tides induced by a tropical storm impinging on a continental shelf', Mon. Weath. Div. 93, 343-355 (1965).

12. Jelesnianski (1972), SPLASH I, 'Landfall storms', NOAA, Tech. M. NWS TDL-46, Washington, pp. 1-52.

13. Myers, V.A. and Malkin, W., 1961, 'Some properties or hurricane wind fields as deduced from trajectories', Nat. Hurr. Res. Report no. 49, Washington, pp. 1-45.

14. T.S. Murty and R.F. Henry (1983), 'Tides in 'the Bay of Bengal', Journalof Geophysical Research, Vol. 88, No. C10, pp. 6069-6076, July 1983.

15. William M. Cray, 1978, 'Recent advances in tropical cyclone research from Ravinsonde composite analysis' - WMO Program on research in tropical cyclones.

16. E.W. Schwiderski, 1979, Global Ocean tides, Naval surface Weapons centre,

KOS,

Dahlgrun, Virginia, 22448.

17. 'Present techniques of tropical storm surge prediction', Report on Marine science Affairs, Report no. 13, World Meteorological

Appendix

CYCLONE PROGRAM DOCUMENTATION

1. Task of the program

Generate a discrete wind stress field for a cyclone, which is given in parametric from.

Parameters for the cyclone history read from the file prepared by the user are: time of observation, latitude and longitude of the eye of the cyclone (both in degrees), pres su re drop (in millibar).

Parameters derived from the above are: maximum wind velocity (in mis), radius of maximum wind (in m).

+

Tl

+---.---+

+

T2 + T3 + T4 + TS

+

-

-

-

---

--

---+

Output are wind stress components on a rectangular, staggered grid. The output is on regular intervals; the input (observations may be irregular.

Tl

I

times of observation T2 T3

I

I

T4

I

TS

I

+---

-

--+---+---+---+---+---+

1<

ST

>1

times of output The procedure is as follows:

1. Cyclone parameters are read from input file;

2. Cyclone parameters are interpolated to output times; 3. Wind shear stress components are computed for the output

CYCLONE program documentation page 2 Hierarchic diagram of subroutines

Main program

I

CYCLONE program documentation page 3 2. Main program

Structure diagram of main program * CYCLONE *

Initialize various constants

Assign values to: ST (time step), MT (number of time steps), SX, SY (mesh sizes), MX, MY (number of grid points).

Read from input file: TCY (time), AL AT (latitude), ALONG (longitude), OP2 (pressure drop);

Call LLTOXY (transforms to Cartesian coordinates); result is XP2 and YP2;

Make: TP2=TCY; For KT = 0 to MT do

Make: TIME = TCY + KT * ST If TIME > TP2

Then TPl=TP2 ; XPl=XP2 ; YPl=YP2 ; OPl=OP2 ;

Read from input file: TP2, ALAT, ALONG, OP2 ; Call LLTOXY (transforms to XP2 and YP2) Interpolate values of XP, YP and OP between TPI and TP2, to time TIME.

Calculate VMAX (max. wind velocity) and RR (radius of max. wind) from OP;

For IX = 1 to MX do For IY = 1 to MY do

Make: X = (IX-O.5)*SX RX = X-XP ; RY = Y-YP R

=

Sqrt(RX**2+RY**2)

Calculate V (velocity) at radius R Calculate TAU (stress)

Make: TAUX(IY) = -TAU*RY/R

Y

=

(IY-l)*SY

Write to file: TAUX(l) to TAUX(MY)

For IX

=

1 to MX do For IY = 1 to MY do

Make: X

=

(IX-l)*SX Y

=

(IY-O.5)*SY RX

=

X-XP ; RY

=

Y-YP

R

=

Sqrt(RX**2+RY**2)

Calculate V (velocity) at radius R Calculate TAU (stress)

Make: TAUY(IY) = TAU*RX/R Write to file: TAUY(l) to TAUY(MY)

The following formulae are used in the program: Vmax = A * Sqrt(OP)

where Vmax is the maximum wind velocity, A a constant (called AVMAX in the program) and OP is the pressure drop. The radius of maximum wind is calculated by means of a

CYCLONE program documentation page 4 linear relationship with

line through a set of rendered graphically:

Vmax, which is found by fitting a observations; the relation can be

Rmax

I

I

RmO

+

I.

I

I

I

I

I

I

I

I

+---+---

> Vmax VmO

The values of RmO and VmO are initial values are: VmO=105 miles) •

to be chosen by (knots), RmO=90

the user; (nautical

The velocities in points anywhere in the calculated by the formula:

field are

v

= Vmax * (r/Rmax)**1.5 if r <= Rmax

v

= Vmax / Sqrt(r/Rmax) if r > Rmax where r is the distance af an arbitrary point to the eye of

the cyclone.

The wind shear stress on the water surface is calculated by: S = B

*

[v]

*

v

where S is the shear stress divided by the density of the water, B is a constant (called AVTAU in the program), which is the product of a dimensionless constant and the ratio of the densities of air and water, and V is the wind velocity defined above. Initial value: AVTAU=O.0028 •

The stress vector is perpendicular to the line connecting the point and the eye of the cyclone. So the components are computed by:

9X = - S

*

y / r Sy = S

*

x / r

CYCLONE program documentation page 5

~<~...

+y

-,

S

,r

-,

"

I

'

"

I

+---+ x

o

(eye of cyclone) 3. Subroutine LLTOXY

Task: transforms latitude and longitude of a point on earth (in this case the eye of the cyclone) into Cartesian coordinates X and Y with respect to grid coordinates.

Parameters: ALAT (I) ALONG (I) X (0) Y (0) latitude longitude

grid coordinates of the point

,

,

4. Meaning of parameters

number of grid points in X-direction number of grid points in Y-direction number of time steps in output

counter of grid points in X-direction counter of grid points in Y-direction counter of time steps

MX MY MT IX IY KT SX, SY mesh sizes

ST time step in output (hours) XLEN length of grid in X-direction YLEN length of grid in Y-direction

TIME time at which output is desired (hours) X, Y coordinates of output point

TP1, TP2 times at which cyclone parameters are given (hours)

ALAT latitude of eye of cyclone ALONG longitude of eye of cyclone ALATl, ALAT2 boundary of grid ALONGl, ALONG2 boundary of grid

RR radius of maximum wind velocity,

DP pressure drop of cyclone

RX, RY coordinates of output point with respect to eye of cyclone R AKNMS DEGRM AVMAX AVTAU

distance between output point and eye converts knots into mis; value 0.51444

converts latitude and longitude into meters; value 1l.E4

constant appearing in formula for maximum wind velocity; va1ue 14.2; pressure drop in millibar, velocity in knots.

initial va1ue is

CYCLONE program documentation page 6

.2..

Input ~ the user

The user must enter the following data into the program: Mx, MY, MT, ST, ALONG1, ALONG2, ALATl, ALAT2

He also must prepare a file containing the following data: time, latitude of eye, longitude of eye, press.drop time, latitude of eye, longitude of eye, press.drop

Souree text CYCLONE model

C PROGRAM CYCLONE

C CONVERTS PARAMETERS FOR CYCLONE INTO WIND STRESS FIELDS

C

PARAMETER (MYMAX=lOO, AKNMS=0.51444, AVMAX=14.2, & DEGRM=11.E4, AVTAU=2.8E-3, AMILEM=1852.)

DIMENSION TAUX(MYMAX), TAUY(MYMAX) INTEGER OUTP

COMMON /MAP/ ALONG1, ALONG2, ALAT1, ALAT2, XLEN, YLEN

C

C ASSIGN VALUES TO VARIOUS CONSTANTS INFILE = 11 OUTP = 12 MX

=

41 MY = 31 MT = 16 ST = 3. ALATl = 26. ALAT2 = 11. ALONGl = 56. ALONG2 = 76. RMO = AMILEM * 90 VMO = 105 *AKNMS C

XLEN = DEGRM * ABS(ALONG2-ALONG1) YLEN = DEGRM * ABS(ALAT2-ALAT1) SX = XLEN / (MX-l)

SY = YLEN / (MY-l)

WRITE (6, 8) ALATl, ALAT2, ALONG1, ALONG2,.XLEN, YLEN, SX, SY 8 FORMAT (' GRID " 9E12.4)

C

C READ FIRST VALUES FROM INPUT FILE READ (INFILE, * ) TCY, ALAT, ALONG, DP2 CALL LLTOXY (ALONG,ALAT,XP2,YP2)

TP2

=

TCY TPl

=

TP2 - l. XPl = XP2 YPl = YP2 DPl = DP2 DO 700 KT=O, MT TIME = TCY + KT*ST IF (TIME.GT.TP2) THEN TPl = TP2 XPI = XP2 YPl

=

YP2 DPl = DP2

READ (INFILE, *, END=800) TP2, ALAT, ALONG, DP2 CALL LLTOXY (ALONG,ALAT,XP2,YP2)

ENDIF R2 = (TIME-TPl)/(TP2-TP1) Rl = l.-R2 XP = Rl*XPl + R2*XP2 YP = Rl*YPl + R2*YP2 RR = Rl*RRl + R2*RR2 OP = Rl*DPl + R2*DP2 = AVMAX * SQRT(DP)

PAGE 2 VMAX = VMAX *AKNMS

RR = RMO * (VMO-VMAX) / VMO WRITE (6, 12) TIME, XP, YP, RR, FORMAT (' CYCLONE PARAMETERS' DO 200 IX=l, MX DO 180 IY=l, MY X = (IX-0.5)*SX Y = (IY-l. )*SY RX = X - XP RY = Y - YP R = SQRT(RX*RX+RY*RY) V = VEL(VMAX, R, RR) TAU = AVTAU * ABS(V) * V TAUX(IY) = -TAU* RY/R

180 CONTINUE

12

DP, VMAX 8E12.4)

WRITE (OUTP, '(6(lX,E12.5»') (TAUX(IY), IY=l,MY)

200 CONTINUE DO 300 IX=l, MX DO 280 IY=l, MY X = (IX-l. )*SX Y = (IY-0.5)*SY RX = X - XP RY = Y - YP R = SQRT(RX*RX+RY*RY) V = VEL (VMAX , R, RR) TAU = AVTAU * ABS (V) * V TAUY(IY) = TAU* RX/R

280 CONTINUE

WRITE (OUTP, '(6(lX,E12.5»') (TAUY(IY), IY=l,MY) 300 CONTINUE

700 CONTINUE 800 STOP

END C

SUBROUTINE LLTOXY (ALONG, ALAT , X, Y) PARAMETER (DEGRM=11.E4)

COMMON /MAP/ ALONG1, ALONG2, ALAT1, ALAT2, XLEN, YLEN IF(ALONG2.GT.ALONG1)THEN

X1=DEGRM*(ALONG2-ALONG) C RIGHT ORIENTED COASTLINE

X=(XLEN-X1) ELSE

C LEFT ORIENTED COASTLINE X=DEGRM*(ALONG1-ALONG) ENDIF IF(ALAT1.GT.ALAT2)THEN Y=DEGRM*(ALAT1-ALAT) ELSE Y1=DEGRM*(ALAT2-ALAT1) Y=(YLEN-Yl) ENDIF WRITE (6, 22) ALONG,ALAT,X,Y 22 FORMAT (' LLTOXY

I,

4E12.4)

RETURN END

FUNCTION VEL (VMAX, R, RMAX) AR = R / RMAX

IF (AR.LE.l.) THEN

VEL = VMAX / SQRT(AR) ENDIF

RETURN END

Nr.

List of figures

1. Test result on .inundation procedure af ter 1600 seconds of model integration.

2. North Sea model area.

3. Bathymetry used for the computation of tides in the North Sea. 4-8. Duchess model simulation results on velocity vectors of the North

Sea using constant bottom at 28000, 44000, 60000, 84000 and 100000 seconds of integration.

9. Sea level variation in relation to time at the point A in the North Sea using realistic bathymetry.

10-16a. Simulation results of tide, flux and velocity vectors of the North Sea at 12000, 44000, 60000 and 84000 seconds of

integration. 16b. 17. 18-23. 24. 25-27.

Bathymetry used for the computation of tide in the Bay of Bengal. M2 tide variation in relation to time at the point B in the Bay of Bengal.

Simulation results of tide, flux and velocity vectors of the Arabian Sea with a constant bottom of 1000 meters after 18000, 34000 and 62000 seconds of integration.

M2 tide variation in relation to time at the point C in the Arabian Sea (ref. figure 18).

Simulation results of velocity vectors and surge after 200 and 1000 seconds of integration.

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