Iceberg drift affected by wave action

Ocean Engng, Vol. 9, No 5, pp.433-439, 1982. Printed in Great Britain.

0029-8018/82/005 0433-01 $03.00/0 Pergamon Press Ltd.

I C E B E R G DRIFT AFFECTED BY W A V E A C T I O N C . C . HSlUNG a n d A . F . ABOUL-AZM Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John's, Newfoundland, Canada AIB 3X5 Al~'tt-~'t--A mathematical model to predict iceberg drift pattern has been developed, which includes the wave drift force, in addition to the other conventional force components such as forces due to wind. current, Coriolis effect, and geostrophic effect. Trajectories of two icebergs were computed first with the wave effect then without the wave effect. All were compared with the observed results from the field. The model with the wave effect shows a significant improvement in the correlation.

1. INTRODUCTION THE RECENTdiscovery of hydrocarbons in the Grand Bank area near Newfoundland and the subsequent tests indicate that this will be one of the producing fields on the Canadian Eastern Seaboard. But there are many hazardous problems that will be encountered for offshore operations in this region. One of them is the presence of drifting icebergs. The possibility of a direct collision of icebergs with production platforms and the scouring of the seafloor by large icebergs are the potential threats. A proper and realistic prediction of the path of a drifting iceberg is a necessary step in the operation of all types of drilling platforms. The trajectory of iceberg drift can be predicted from mathematical models based on information about the iceberg characteristics and about the environmental forces affecting its motion. Mathematical models for predicting drift trajectories have been developed among others, by Cochkanoff et al. (1971), Sodhi and Dempster (1975), Napoleoni (1979), Sodhi and EI-Tahan (1980), Mountain (1980). The environmental forces considered were the Coriolis force, current force, wind force, and the force due to geostrophic water currents. However, the wave force has always been neglected in these drift models. In this paper, an iceberg drift model is presented to include the action of ocean waves. Predicted trajectories based on the mathematical models both with and without wave effect are compared with the observed results from the field. 2.

WAVE DRIFT FORCE

The wave forces acting on a floating body may be divided into two parts: (a)

(b)

First-order wave force: it is linearly proportional to the wave amplitude having the same frequencies as the waves. It is periodic in nature with a zero mean for time average. Second-order wave force: it is a time averaging and slowly-varying force with frequencies below wave frequencies. Its magnitude is proportional to the square of the wave amplitude. 433

434

C . C . HSIUNG a n d A . F . ABOUL-AZM

The first-order wave force is the so-called periodic wave exciting force which is responsible for the motions of floating bodies with wave frequencies and predominant in wave loading on an ocean structure. However, it is the second-order wave force which causes the free floating body drifting away from its initial position. Thus the second-order wave force, namely the wave drift force, is also important in the mooring system analysis for a floating body in waves. Wave drift force on an iceberg has been computed with the wave diffraction theory using a three-dimensional singularity distribution on the wetted surface of the floating body in regular waves. Similar methods were developed by Faltinsen and Michelsen (1974), and Pinkster and van Oortmerssen (1977) to compute the drift force for floating structures. However, in the case of an iceberg, its velocity and oscillating motion are very small. It is assumed that the iceberg is "fixed" in the free surface. Hence the radiation effect due to the oscillatory motion can be neglected. The total fluid velocity potential is assumed to take the form: + = +~ + +E,

(1)

where Ct is the velocity potential of the incident wave, and 6 0 is the wave diffraction potential. From Bernoulli's equation the pressure on a floating body is ado at

P = -P

where p g z

1 2 p Iv¢1)12 -- pgz

(2)

= fluid density, = gravitational acceleration, = vertical coordinate which is zero at the undistributed free surface and positive upwards.

The wave elevation according to the linearized wave theory is =

1 Oqb g

for z = 0 .

(3)

Ot

Finally the wave drift force is the time averaging the integration of the pressure over the wetted surface of the floating body. It can be written in the following vector notation: T

1 I dt{- f sfpndS} T

Fwv

O

T

=--

dt - -

T o

pg

~2nds L

Iceberg drift affected by wave action

435

1.00: .d

H >. 200 rn - - - . - - H = lOOm

WATER DEPTH

0.8( U.

u.

0.60

%

n-

O 0

0.40 o o z

0.20 -

0.00 10.00

I

I

I

I

12.00

1400

16.00

18.00

2000

ENCOUNTERED WAVE PERIOD (SECONDS)

FIG. 1.

Wave drifting force factor for a 200,000-ton tabular iceberg (90 x 90 × 25.5 m).

+ f s f ~-p where T t n

S

WL

= = = = =

IV~b]2 ndS}

(4)

wave period of encounter, time, outward normal vector to the surface, wetted surface, water line.

In regular waves, the drift force is steady and its magnitude is constant depending on the wave amplitude. The wave drift force on a typical medium size tabular iceberg. 90 × 90 x 25.5 m weighing 200,000 tons has been computed. The results are shown in Figs 1 and 2. 3.

THE M A T H E M A T I C A L MODEL OF ICEBERG DRIFT

We only consider the translatory motion of an iceberg in a horizontal xy-plane. Since the oscillatory motion of the iceberg is assumed to be very small, the added mass due to local disturbances in the neighbourhood of a floating body and the damping effect due to radiated waves o f the body motion can be neglected. There is no restoring forces in the xand y-directions. The equation of motion of iceberg drift can be written in the following vector notation: Ma = F.,,, + FA + Fw + Fc + Fc,

(5)

436

C.C. HSlUNG and A.F. AaOUL-AZM 500

IS WAVE PERIOD IN SECONDS

%

T 12 =14

400 =

z o

3.00

g u_

2.00 u_

g uJ >

1.00

000 000

I LO0

200

300

WAVE AMPLITUDE

FIG. 2.

where

4.00

5 O0

(METERS)

Wave drifting force acting on a 200,O00-ton tabular iceberg (90 × 90 x 25.5 m).

M a Fw,, FA Fw Fc F(;

= = = = = = =

iceberg mass, acceleration, w a v e drift force, air drag due to wind, water drag due to current, Coriolis force, force due to geostrophic current.

T h e equation of m o t i o n is m o r e explicit in a c o m p o n e n t form as follows:

Mddt,

1 p,~4a ]y.]2 sin 0 = (F.,,,).~ + --~-Ca +

1 ~

Cwp~~ m , (uci-u) Iv,,-vb

+ Mfv-

(6)

Mfve = "£.F~ .

MdV 1 dt = (Fw,,)y + 2 - C, paA,, tV,,I 2 cos 0 1

+ 2 Cwpw ~ Aw, ( v d - v ) - Mfu + Mfu~ = Y~Fv

Iv,,-vl (7)

Iceberg drift affected by wave action

437

where V

=

Va

=

Vcj

=

Vg

=

Cw

=

Pa

=

PW

0

=

m a

=

Awj

=

f

=

(u, v), iceberg velocity, (u~, v~), wind velocity, (u ovcj), water current velocity at jth layer of iceberg under water, (ug, vg), geostrophic current velocity, air drag coefficient, water drag coefficient, air density, water density, wind direction, area of iceberg profile above water perpendicular to wind direction, area of iceberg profile of jth layer under water perpendicular to current direction, 2 l-I sin 0, Coriolis parameter with fl angular velocity of earth and latitude location of iceberg.

Except the wave drift force term, the other force terms have been discussed by Mountain (1980), and Sodhi and EI-Tahan (1980) in great detail. Our interest is to see how the additional force term of the wave drift force will affect the iceberg drift pattern. To determine the iceberg trajectory we solve the following initial value problem numerically dx dt

--

U

(8)

.

dy dt - v .

(9)

d u = _1 E F, dt M " "

(10)

dy _ 1 EF,, dt M - "

(1t)

For a given initial condition of the iceberg, the above set of equations are stepped forward in time to yield predicted positions. The values at time t + At are obtained simply by the first-order Taylor series expansion u(tt + At) = u ( q ) + A t . it(h)

(12) v(t, + At) = v(h ) + A t . i'(q) .

Since the acceleration u, i) of an iceberg vary only slowly with time, this method should be able to provide sufficiently accurate solutions. In this work the time step is taken to be one hour. 4.

NUMERICAL EXAMPLES

Two case examples have been selected to compute the predicted drift trajectory first with the wave effect then without the wave effect. All are compared with the observed results. They are shown in Figs 3 and 4.

438

C.C. HSIUNGand A.F. ABOUL-AZM t5.00 OBSERVED

z - -

I

I0.0(

-'~,.~

ICEBERG TRACK

---

PREDICTED

TRACK

WITHOUT

--.

PREDICTED

TRACK

WITH

WAVE FORCE

WAVE DRIFT

FORCE

g z

(2:

5.0(

O I-u_ @

O.OO

- 5.0C -900

I -6.00

I -3.00

I 0.00

DRIFT

FIG. 3.

TOWARDS

1 3.00 EAST

t 6.00

I " ~"i 9.00

12.00

(N.M.)

Drift trajectory of iceberg G181.

Iceberg drift data were taken from Appendix 3 of Offshore Labrador Environmental Conditions, Summer 1974. Two icebergs, G181 and G183, were chosen. Their estimated weights were in the order of 200,000 tons with a tabular shape, and the data for these two bergs were complete with the details of current, wind and waves. 5.

CONCLUDING REMARKS

From Figs 3 and 4, it is obvious that the wave effect consideration in the mathematical model has been improved significantly the predicted iceberg drift trajectory. From

iO00

z

OBSERVED

500 3=

g

~:~

z

_~

"'C\.~___

"i'

OOC

ICEBERG TRACK

PREDICTED TRACK W I T H O U T WAVE FORCE PREDICTED

"%

TRACK

WiTH

WAVE DRIFT FORCE

-5 O 0 o

-,OOO -700

l -5.o0

I -3oo DRIFT

FIG. 4.

I -~.oo TOWARDS

t ~oo EAST

I 3,oo

(N.M.)

Drift trajectory of iceberg G183.

I 5o0

T.O

Iceberg drift affected by wave action TABLE 1.

439

COMPARISONBETWEENTHE MAGNITUDEOF DIFFERENTENVIRONMENTALFORCESACTINGON A 200,000-TON TABULARICEBERG

Force components

Force (tons)

Water drag

5.35

Wind drag

1.95

Coriolis effect

0.76

Geostropic effect

4.0

Wave drift effect

8.11

Condition

Force (tons)

Condition

Relative vel. = 0.2 m/sec Wind speed = 20 knots Latitude x 55 ° Rel. vel. = 0.2 m/sec Acceleration = 2 x 10 -5 m/sec 2

15.12

Relative vel. = 0.4 m/sec 7.8 Wind speed = 40 knots 1.52 Latitude = 55 ° Rei. vel. = 0.4 m/sec 8.0 A c c e l e r a t i o n = 4 x 10 -5 m/sec 2

Wave amp. = 0.5m Wave period = 12 sec

32.45 Wave amp: = 1.0m Wave period = 12 sec

Force (tons) 34.02

Condition Relative vel.

17.55 Wind speed = 60 knots 2.27 Latitude = 55 ° Rel. vel. = 0.6 m/sec 12.0 Acceleration 6 × 10 -5 m/sec 2 73.00 Wave amp. = 1.5m Wave period = 12 sec

T a b l e 1, we find that t h e m a g n i t u d e o f the w a v e d r i f t f o r c e is in t h e s a m e o r d e r o f t h e m a g n i t u d e o f t h e s u m o f o t h e r force c o m p o n e n t s for a 200,000-ton t a b u l a r i c e b e r g . A s a c o n t i n u a t i o n o f this s t u d y , t h e size effect a n d s h a p e effect o n t h e w a v e d r i v e f o r c e a r e at p r e s e n t u n d e r e x a m i n a t i o n . H o w e v e r , it is n e c e s s a r y f o r a n y m a t h e m a t i c a l m o d e l l i n g o f i c e b e r g drift to i n c l u d e t h e w a v e effect, p a r t i c u l a r l y f o r s m a l l a n d m e d i u m size i c e b e r g s .

Acknowledgement-- The authors acknowledge the support of the Natural Sciences and Engineering Research Council of Canada in funding this project. REFERENCES Cocnr~NOFF, O., GRAHAM, J.W. and W ~ S E r , J.L.1971. Simulation techniques in the prediction of iceberg motion. Proc. Can. Semin. Icebergs, Halifax, Nova Scotia, pp. 135-152. FALnNSE~, O.M. and MXCnELSEN, F. 1974. Motions of large structures in waves at zero Froude number. Proc. Int. Symp. Dynamics of Marine Vehicles and Structures in Waves, London, pp. 91-106. MouwrA[N, D.G. 1980. On predicting iceberg drift. Cold Reg. Sci. Technol. 1,273-282. NAPOLEONI, J.G. 1979. The dynamics of iceberg drift. M.Sc. Thesis, Dept of Geophysics and Astronomy, Univ. of British Columbia. Offshore Labrador Environmental Conditions. Summer 1974. Marine Environmental Service Limited, Calgary, Canada. PINKSTER, J.A. and VAN OORTMERSSEN, G. 1977. Computation of the first and second order wave forces on bodies oscillating in regular waves. Proc. 2nd Int. Conf. on Numerical Ship Hydrodynamics, Univ. of California, Berkeley, pp. 136-156. Soom, D.S. and DEMPSTEB, R.T. 1975. Motion of icebergs dlie to changes in water currents. IEEE Oceans 75, 348-350. SODm, D.S. and EL-TAHAN, M. 1980. Prediction of an iceberg drift trajectory during a storm. Z. Gletscherk. Eiszei(forsc. Gesh. Klimas. 1, 77-82.