A note on surface wind-driven flow

Ocean Engng. Vol. 6, pp. 273-284 Pergamon Press Ltd. 1979. Printed in Great Britain

A NOTE ON SURFACE WIND-DRIVEN FLOW JOSEPH M. BISHOP Environmental Data and Information Service, National Oceanic and Atmospheric Administration, Washington, D.C. 20235, U.S.A.

Abs'traet--A simple operationally oriented model of surface wind-driven currents is presented in which Lagrangian surface drift is assumed to be composed of a linear combination of a wave-induced Stokes drift plus a wind-driven Ekman drift. Using this approach, Stokes drift accounts for as much as half the total surface current magnitude. The Lagrangian current is predicted to be about 3.5~ of the 10 m wind magnitude directed in the sense of an Ekman spiral about a 20o deviation angle. For comparison to this model, a second model is proposed that accounts for the interaction of Stokes current and Coriolis force. An inference drawn from this model is that there is only weak coupling between Coriolis force and Stokes drift. Such a conclusion, if correct, leads one to focus attention on the Lagrangian model for operationally oriented current estimates. Results of the Lagrangian model agree with observations of investigators for currents at the air-sea interface and may have application in the movement of oil slicks or surface drifters at sea .under fetch or duration limited sea states. INTRODUCTION

MANY variations o f Ekman's (1905) original work have been developed, but none have included the joint calculation o f transient wave and wind induced flow. It is well known that the sum o f the Stokes and Eulerian currents equals the Lagrangian current (LonguetHiggins, 1969). Ursell (1950) showed that in theory this Lagrangian current vanishes for fully developed seas on a rotating earth. Iaaniello and Garvine (1975) pointed out that either in fetch orduration limited seas this current does not necessarilyvanish and is directly useful in circulation studies. Estimates o f the spectral form o f wind-waves may be made from meteorological inputs using available models or by remote radar techniques. With such a specification o f wave spectrum, one may make an estimate of Stokes currents and the friction vel¢ city. The shear stress, eddy viscosity coefficient, and subsequently the Ekman current can be estimated given the friction velocity. Such wave field related estimates are the basis o f this note and are important to many ocean engineering applications. S P E C I F I C A T I O N OF THE G R O W I N G WAVE SPECTRUM The short-crested sea surface may be characterized theoretically by an "energy density" spectrum, S(co), where co is the radian frequency of a long-crested Fourier component of the wave field. A generally accepted wave generation theory is the Inoue (1967) model (Dexter, 1974). In Inoue's model, growth continues until a predetermined mathematical cut-off has been reached. A convenient formula for this limit was proposed in PiersonMoskowitz (1964) as

$'®(tO) = ag~2exp

~5

[

273

-- [~

(1)

274

JOSEPH M. BISHOP

where ~ = 8.1 x 10-a, ~ = 0.74, coo -- ,a,v-~5 9 Vxg.5 is the wind measured at 19.5 m, and g = 980 cm see-*-. Inoue's governing equation for energy density is given as dS dt

A 1 --

+ BS \-~/

1 --

J

(2)

where A is a linear growth factor, B is an exponential growth factor, and t is wind duration. The solution to equation (2) for zero initial spectrum and infinite fetch was given by Inoue as

S(o~,t)=[A(exp_~Bt)--l)][l-l-.ffff_~ I)}'] -I. ~ A(exp(Bt) --

(3)

Obviously, the results presented in this note depend on the accuracy of the spectral model employed. Ianniello and Garvine (1975) show that the Inoue model is reasonably suited for calculation of Stokes currents in a growing sea state. STOKES DRIFT IN A RANDOM WAVE FIELD The forward motion of fluid particles in an irrotational gravity wave is not completely compensated by the backward motion. Thus, a mean Stokes drift in the direction of wave propagation (more or less downwind in a growing sea) is predicted. Attempts have been made to theoretically estimate Stokes drift (e.g. Bye, 1967; Chin, 1971; Kenyon, 1969,1970). The results of such studies indicate that surface wave-induced currents are comparable in magnitude to Ekman surface drift. For a statistically stationary and horizontally homogeneous random gravity wave field, Keynon (1969) calculated a downwind Stokes drift velocity, V,, given in this note for the special case of a uni-direetional spectrum as

s(co)o,exp [(m,z)/gldo

v, (z) =

(4)

where Z is the vertical coordinate. Kenyon (1969) also calculated a maximum value for surface Stokes drift by letting S(co) -- S**(co) in equation (4) such that a Stokes drift wind factor was given as V,(o) _ 1.57%.

(5)

~'/19 • 5

He further estimated that, by neglecting the angular spreading of the waves in this calculation, he overestimated this factor by about 10 %. The calculation of a transient Stokes current in a growing wave field has been accomplished by computer methods (Chin, 1971). In this analysis surface Stokes drift was

A note o n surface wind-driven flow

estimated by a numerical summation such that equation

V.(o, t) = _2

s(e,.+,, 0

275

is approximated as

(4)

Ae

(6)

g ,'.°0.038 r - - 5 00 where r = the number of spectral components, Ac0 = c0~,+x--oh, = 2n A f In this note, we take A f = 0.006 sec-1 and estimate S(o)~+~, t) from the Inoue model. Example calculations of surface Stokes drift with 19.5 m winds of V]9.5 -~ 1799 crn sec-1 (35 knots) and Vlg.s = 2056 cm sec-; (40 knots) using this summation technique, are presented in Fig. I. It is interesting to note the close agreement between these calculations at steady state and Kenyon's fully developed value of 1.57 %. Repeated numerical summations using equation (6) with smaller values of Ac0 and larger values for r lead to even closer agreement with Kenyon's value. Although this irrotational Stokes current model seems to give reasonable values of wave-induced current magnitudes, fluid motions at sea cannot be truly irrotational. Regardless, limited laboratory results (i.e. Russell and Osorio, 1958) do indicate that the surface drift predicted by Stokes' inviscid theory is a good estimate for deep water. At this point, it is not apparent as to the extent that irrotational theory holds in an active wave field, but to take advantage of the apparent success of potential theory in describing the dynamics of gravity waves, the irrotational formulation can be used as a starting point. Another assumption implicit in this Stokes formulation is that Coriolis influence can be neglected in the calculation of wave induced currents. Although the influence of Coriolis force on duration limited seas (i.e. growth periods less than inertial) is minor, it may actually affect drift of sufficient duration. One possibility is that once a Stokes drift is established, Coriolis force would cause it to rotate and Stokes momentum would be transferred to an inertial

~ / ~ l ~" t.57% 0.015

\

0.0125

.

4-

$

0.010 ~ 5= 1799 cm see-' =

0.00?5

1(9+5=2056 ¢rn sec -b . . . . . .

0.005

I

FO

I

20

I

~0

t

40 Duration,

I

I

~

~0

I

7"0

I

80

I

hr

FIo. 1. E x a m p l e calculations o f n o r m a l i z e d Stokes drift ( V , / V l , . s ) u s i n g 19.5 m winds, V,.~, o f 1799 can sec -x a n d 2056 crn sec -1 a s a f u n c t i o n o f w i n d d u r a t i o n . T h e calculation used 500 spectral c o m p o n e n t s between frequencies 0.038 sec -1 to 3.038 sec -x with A f := 0.006 sec-L

276

JOSEPHM. BISHOP

type of oscillation (Chin, 1971). Following other simple formulations of this problem (i.e. Bye, 1967; Kenyon, 1969, 1970), such considerations of the influence of the viscous and Coriolis forces, although of possible importance, will not be fully addressed. TRANSIENT EKMAN WIND-DRIFT MODEL The second component in the proposed surface drift model is the current driven by surface shear stress. The governing Ekman equation for the complex velocity, We = U, + iV,, is given as __

02We

We + i t We -- , -- 0 Ot OZ"z

(7)

where ~ is the Coriolis parameter, e the uniform eddy coefficient, and i = (-- 1)~. U, and V, are the x and y-component Ekman velocities, respectively. Equation (7) is subject to the boundary conditions s 0 W, _ *x(t) + i*j,(t) _ z(t) ; Z = 0 ~Z p p W, = O ; Z

(8)

= -- oo

where %,(0 is the x and zy(t) the y stress component, respectively and p is the water density. When the condition of initial fluid rest, i.e. W , = O ; t <_O is assumed, the solution found by Fredholm as reported by Ekman (1905) is W,(Z, t) -- ~

z(t -- )~exp[-- {i ~ ~ + (Z~/4e~)}] d~

f'

p(~n)i 0

(9)

(~F

in which ~ is a d u m m y variableof integration. For a constant shear stress,surfacecurrentsmay be expressed in component form as

U,(o, t) = ~

and

V,(o, t) --

1

1

p(sx)~

[ ~ w + ~yv,1

(10)

[-- Zx72 + Zyy1].

The weighting functions ~'z and T~ can bc detcrmined by numerical summation of f' c o s ~ 7x = 3o ( ' - ~ d ~ and

=

• fi~, X

c o s ~ d ~ , J = 32

A note on surface wind-driven flow

=

(*sinf~dE

=

277

t ''~r sinf2~ u-,Ju-,)~, C~) ~

where At has been arbitrarily taken as 3 hr and t taken as 96 hr in this note. Although the problem of a linearly dependent eddy coefficient under transient forcing has been addressed (e.g. Madsen, 1977), the details of the depth dependence of eddy viscosity have not been completely documented to date. Because of this difficulty, only a uniform vertical exchange coefficient, z, will be considered in this calculation. For this application, the crude semiempirical formula for ~ given by Csanady (1976) is considered adquate, in which --

pa V.z

200pf~

[c.g.s.].

(11)

pa is the air density and V. is the friction velocity. The friction velocity can be estimated from the modelled wave field using the empirical formula of Hsu (1974). SURFACE DRIFT MODEL FOR A CONSTANT UNI-DIRECTIONAL WIND Using the theoretical work of Longuet-Higgins 0969) as a basis (i.e. Lagrangian current ----- Stokes current + Eulerian curren0, it is assumed that the transient Lagrangian winddriven current is composed of a downwind Stokes current plus a Eulerian current. Justification for adding a non-linear Stokes current to the linear Ekman flow is not specifically addressed in this note. One is led to this superposition basically because the model seems to fit the limited observations, as will be discussed later. Given a constant wind, the Stokes current continually grows with duration time to about 1.5~o of the wind speed (e.g. Fig. 1). For these same conditions, Ekman currents behave as in the commonly theorized Ekman spiral. The friction velocity is computed, as is the Stokes current, from the growing spectral wave field using the Inoue model. Viscosity coefficient, ~, and surface shear stress, ¢ ---- paV.L are estimated from the friction velocity. In this manner, theoretical surface drift currents, W, for a wind blowing along the positive y-axis are given as W = (Do2 + VoS)i

(12)

with

Vo=V,+V, and the deflection angle, 0, between wind and surface current defined as 0 =

tan-~CUolVo).

Using this model, the response to a spatially homogeneous wind was examined for two cases: V10 = 1028 cm sec-1 (20 knots) and V10 = 1542 cm sec -1 (30 knots). The results of

278

JOSEPHM. BISHOP 6.0

/

1

50 L -

~0" 1028 cm SeC-I

' I

__~ 10.6

--=k=W/V~o

/

4ol-

_

r \

I"

. . -

Tp

/ -1

"+

-Io.~

.-..

.

T ++`

I0

OZ

o

6

Iz

m

24

3o

36

42

48

~4

Wind duration t,

6o

66

zz

78

84

9o

hr

FIG. 2. Example calculations of wind factor (in perccrit) k = W/Vlo and Stokes drift divide.d by total drift (V,/W)usi~q~ a 10 m wind, V=e,of 1028 crn scc-1 as a function of wind duration, t, in hr for latitude 45 °. Also, T p = 2~/Q is the inertial period.

this calculation are shown, (Figs 2 and 3), where values o f wind factor k (i.e. W/Vlo) and the ratio o f Stokes drift to total drift (i.e. VJW) are given. As indicated, b o t h o f these factors increase rapidly a n d a p p r o a c h a steady value with k - 3.5% and V,/W = 0.45. The E k m a n c o m p o n e n t causes these factors to undergo a d a m p e d oscillation a.bout the steady-state value with the inertial period (Tp = 2¢/f~). Figures 4 and 5 are h o d o g r a p h s that

I~'o-1542 cm sec-~ - - ='k =W/V~o ---- = VJW

5.o

1 0.6

"-6 ~- 4.0

0.5 .i-

II

3.0 ~2

.g 7

2.0

I0

--0.3

Oz

--

I

6

I

12

I

18

I

24

I

30

I

36

I

42

J

48

Wind duration t,

I

54

I

60

I

66

I

72

1

78

l

84

t

90

"~

(/3

+

hr

FIG. 3. Example calculations of wind factor (in percent) k = IV/I11oarid Stokes drift divided by total drift ( V,/ W) using a 10 m wind, V~0,of 1542 crn sec-1 as a function of wind duration, t, iri hr for latitude 45°. Also, Tp = 2r~/fl is the inertial period.

A note on surface wind-driven flow

279

Y 4o~_

Vio~1028 cm sec-I

T

~ ~ " 7~'~.

TO

Pro~sed model

I

2O

/-/o Crosswind surfoce current component,

.X cm sec-~

Flo. 4. Example calculations of surface down-wind, I/,, and cross-wind, Ue, current components using the propused model (dashed line, with numbers in triangles that indicate wind duration in hr) and the classical Ekman 0Fredholm) model (solid line, with numbers in circles that indicate wind duration in hr) for a wind of 1028 cm sec-1 and • given by (11).

indicate the x- and y-components of modeled surface drift for a wind along the y-axis. Note the final deflection angle of about 20°. Such calculated values of k and e are in rough agreement with the collected observations given in Stolzenbach et al. (1977) as shown in Table 1 and indicate a range of k from 0.8 % to 5.8 % with deflection angles between 0 and 13°. Tomczak (1964) further reported that observed values ranged from 1--4~o, with the largest values corresponding to currents measured directly at the sea surface (smaller values measured a few metres below). Such a rapid decrease of k with depth qualitatively agrees with the shear predicted for Stokes drift (Kenyon, 1969) and is thus consistent with this approach. Also, observations of surface oil slicks consistently exhibit values of 0 -~ 10°, in rough agreement with this model and far less than the 45° predicted by Ekman theory (i.e. Smith, 1968; Teeson et al., 1970). These results suggest the importance of Lagrangian surface currents as a possible important component in the movement of oil slicks. A COUPLED STOKES/EKMAN CURRENT MODEL

In the last section a Lagrangian model of surface currents is proposed. Such a model fails to consider the effect of Coriolis force on the Stokes current. This interaction would actually manifest itself as an additional force in the equation of motion (e.g. Kraus, 1972, p. 223). Consider the steady Ekman balance between Coriolis and frictional forces that includes the additional forcing due to the Coriolis-Stokes effect. If the Stokes current is of known magnitude, the steady equation of motion becomes

280

JOSEPH M. BISHOP

60 1

Vlo- 1542 ¢m $~-I Proposed theory

t--

~, //

~Fr~lholrn

3o

I0 20 30 40 /.,t o Crosswind surfQ~ current component,

X cm sec-~

FIG. 5. Example calculations of surface down-wind, 11o,and cross-wind, Uo, current components using the proposed model (dashed line, with numbers in triangles that indicate wind

duration in hr) and the classical Ekman (Fredholm) model (solid line, with numbers in circles that indicate wind duration in hr) for a wind of 1542 cm sec-x and • given by (11).

i~We

--

• azWe -az 2

(13)

iflV s

in which the Stokes current, progress downwind along the y-axis and V, is given in (4). Although V, is generally a function of depth, the irrotational nature of Stokes drift does not allow the introduction of a viscous term analogous to • a~W'e[aZ2 for Vs in equation (13). The solution to the governing differential equation can be expressed in a form that resembles Ekman's (1905) solution, with additional terms that account for the interaction of Coriolis force and Stokes current. The result, a coupled Stokes--Ekman model, can be expressed in terms of a cross-wind component, U, given as

u

exp[2O, zqa o (14)

2 ,-m

and a down-wind component, V, given as

\

g

lkf~l

Lab experiment

Hindcasting of observed oil slick Hindcasting of observed oil slick Field experiment with drift cards Field experiment with drift cards Field experiment with drift cards Field experiment with drift cards Field experiment with plastic sheets Field experiment with drifting oil Lab experiments with oil as drift medium Lab experiments (at high Reynolds numbers) Field experiments in a small basin Field experiments made off fixed platform Using a drifting current pole

Nature of experiment 3.4 4.3 4.2 2.1 2.2 4.2 2.8 0.8 3.7 3.3 3.3 1.6 1.2 4.3 5.8 4.1

Mean wind factor (~) 0.7 N.A. N.A. 0.4 0.4 N.A. 1.1 0.7 0.2 none none none none none none 0.9

Standard deviation (~) 3.3 right 0.0 0.0 3.5 right 0.3 left 0.0 13 right none N.A. N.A. N.A. 5.2 right 13.2 right 1.9 right 4.8 right N.A.

Deflection angle (degrees)

SUMMARY OF VARIOUS EXPERIMENTS INVESTIGATING THE EIq[rECT OIF WIND ON SURFACE DRIFT (AIRIER STOLZENBACH e t

Smith (1968) Tomczak ( 1 9 6 4 ) Tomczak (1964) Hughs (1956) Hughs (1956) Nemnann (1966) Teeson et al. (1970) Smith et al. (1974) Swartzberg (1971) Keulegan (1951) Van Dorn (1953) Doebler (1966) Doebler (1966) Doebier (1966) Doebler (1966) Wu (1968)

Name of investiptor

TAaLE 1.

11.0 N.A. N.A. 10.7 8.6 N.A. 7.0 none N.A. N.A. N.A. none none none none N.A.

Standard deviation (degrees)

al., 1977).

I,o OO

O

O

e'L

g 3.

ft O

o

282

JOSEPHM. BISHOP

V = Kexp[(fl/2~)tZ]sin[(fl/2~?Z + ?] +

g

n-0

- exp [2~'IZ]Ac0

Am--O.O3S r ,,,.q)O

g

I\fil

(15)

+I

with S~o (o2~-1) given by equation (1). The constants of integration, K and % are easily determined by applying the boundary conditions

~v

~y

~U--0,~=--;Z=0 aZ c~Z p and

U=V=O,Z=--om

In this manner, it is found that

ry

p~

~

2 ~-0 -- Am--0.038 g ,-soo

+ g

=

[ J

(16)

S®(c0~+0:'o I

(17)

,--7-: I\~I + I

tan-1

2 +

~-0

c0%,+iS®(o~,+0Ao

- ~--o.o3s

' ' ' - g- ' \ I \ ~ I

+I

and

[

K

,-I 2Izy / p(¢ f~)i (cos ? + sin q~J L

2 z ~

3 ~ 3/~

2 n..o -g - ~ - ,.s0o -O.03S + 1

The coupled model is now compared to the steady-state results obtained from the first model. Using values of/:1o = 1028 cm sec-1 (20 knots) and/:10 = 1542 cm sec-1 (30 knots) the steady coupled model gives values for surface wind-factor, k, of 1.8 % and 2.0 % respectively and in both cases a surface deviation angle, % of about 43°. Such results are almost exactly those one would obtain using the Ekman component of the Lagrangian model alone. One might now speculate that the coupling between Coriolis force and Stokes current is of relatively minor importance, and that the Lagrangian model approach cannot be dismissed on these grounds.

A note on surface wind-driven flow

283

CONCLUSIONS An observationally reasonable model of surface current is proposed that combines both Stokes current and a wind-driven Eulerian current. The model predicts that in deep water Stokes drift is an important component of the resultant ocean surface current. The calculation presented was computed for the level Z ---- 0, exactly at the air sea interface, and thus, the results can be used to approximate motions o f such objects as drift cards or may also have application to oil spill trajectory calculations which to date do not include wave current estimates. Predicted surface-drift generally agrees, both in magnitude (i.e. k -~ 3 . 5 ~ ) and direction (i.e. q) - 20°), with observed values. The Lagrangian model proposed extends the steady-state analysis o f Kenyon (1969, 1970) to the transient case, while the coupled model proposed points to the justification o f the Lagrangian model. This note was written to suggest the importance of including estimates of the wave field in problems concerning upper ocean currents. Wave field estimates are available either from wind-wave modeling techniques or possibly using radar remote sensing, and can subsequently be related to Stokes currents, shear stress, eddy viscosity coefficient, and E k m a n currents. The proposed model can also be applied to conditions where the dominant wave pattern is not downwind. Under these conditions one must consider the directionality of the wave frequency spectrum (e.g. Chin, 1971). Such calculations might be useful in accounting for some of the scatter in surface current observations. Finally, although the proposed Lagrangian model seems to predict reasonable surface current magnitudes it, at best, only approximates the actual physics of the problem. A more reasonable approach to the problem might include a better specification of the eddy viscosity (i.e. e(Z, t) in a transient coupled model that includes vertical motion. Such a model is beyond the scope o f this short note which was written with the spirit of forming an operational usable predictive tool for oil trajectory estimates under either fetch or duration limited sea states that fits available observations. Acknowledgements--I wish to thank Fred Everdale of N.O.A.A. and my associates at the U.S. Coast Guard Oceanographic Unit for their helpful comments. REFERENCES BYE, J. A. T. 1967. The wave-drift current. J. mar Res. 25, 95-102.

CmN, H. 1971. An Evaluation of Stokes Velocities and Inertial Currents Generated by Deep-Water Surface Gravity Waves. New York University Geophys. Sci. Lab. Report. 71-11. CSANAVY,G. T. 1976. Mean Circulation in Shallow Seas..7. geophys. Res. Sl, 5389-5399. DEX~R, P. E. 1974. Tests on Some Programmed Numerical Wave Forecast Models..I. phys. Oceanogr. 4, 635-644. DOEeLER,J. J. 1966. A Study of Shallow Water Wind Drift Currents at Two Stations off the East Coast of the U.S.U.S. Navy Underwater Sound Lab., USL Report No. 755, New London, Conn. EI~MAN,V. W. 1905. On the influence of the Earth's rotation on ocean currents. Ark. Mat. Astr. Fys. 2, 1-53. Hsu, S. A. 1974. A Dynamic Roughness Equation and Its Application to Wind Stress Determination at the Air.Sea Interface. J. phys. Oceanogr. 4, 116-120. HuoHs, P. 1956. A Determination of the Relation Between Wind and Sea-Surface Drift. Q. Jl. R. met. Soc. 82, 494-502. INOU~,T. 1967. On the Growth of the Spectrum of a Wind-Generated Sea According to a Modified Miles. Phillips Mechanism and its Application to Wave Forecasting. New York University Geophys. Sci. Lab. Report 67-5. IANNIELLO,J. P. and GARVINE,R. 1975. Stokes Transport by Gravity Waves for Application in Circulation Models. J. geophys. Res. 5, 47-50. KENYON,K. E. 1969. Stokes Drift for Random Gravity Waves. J. geophys. Res. 74, 6991-6994. KENYON,K. E. 1970. Stokes Transport. J. geophys. Res. 75, 1133-1135.

284

JosePH M. BISHOP

KEULEGAN,G. H. 1951. Wind Tides in Small Closed Channels. J. Res. natn. Bur. Stand. 46, Res. Paper 2207, pp. 358-381. KILOS, E. B. 1972. Atmoapheric--Ocean Interaction, Clar~don Press. Oxford, England, pp. 275. LONOUrr-HIQoINS, M. S. 1969. On the Transport of Mass by Time Varying Ocean Currents. Deep Sea Res. 16, 431-447. M ~ a ~ , O. S. 1977. A Realistic Model of the Wind-Induced Ekman_ Boundary Layer. J. phys. Oceanogr. 7, 248-255. NEUM~,'%H. 1966. The Relation Between Wind on Surface Current Derived from Drift Card Investigations. Dr. hydrogr. Z. 19, 253--266. PIBnSON,W. J. and Mo$1cOWlTZ,L. 1964. A Proposed Spectral Form for Fully Developed Wind Seas Based on the Similarity Theory of S. A. Kitaigorodskii. J. geophys. Res. 69, 5181-5190. RO~LL, R. C. and OsoPdo, J. D. 1958. An Experitnental Investigationof Drift Profiles in a Closed Channel. Sixth Conf. Coastal Engineering, Chap. I0, 171-193. SMITH,C. L. and MACIrCrYU,W. G. et al. 1974. Investigations of Surface Films-Chesapeake Bay Entrance. EPA 670/2-73-099, U.S. Government Printing Office, Washington, D.C. S~ITIt, J. E. 1968. Torrey Canyon Pollution and Marine l.z'fe. Cambridge University Press. STOLZm~ACn,K. D., M ~ , O. S., ADAMS,E. E., POLLACK,A. M. and CORTmK. COOFEI~.1977. A Review and Evaluation of Basic Techniques for Predicting the Behavior of Surface Oil Slicks. Massachusetts Institute of Technology Dept. of Civil ~n~mm'ing Rcpt. 222. SwxitTZ~I~O, J. (3. 1971. The Movement of Oil Spills. Prac. of the Jr. Conf. on Prey. and Cont. of Oil Spills, published by Arner. Petrol. Inst., Washington, D.C., pp. 489--494. T~ESON,D., WHITE,F. M. and SCrmNCK,H. 1970. Studies of the Simulation of Drifting Oil by Polyethylene Sheets. Ocean Engng 2, 1-11. ToMcz.t~ G. 1964. Investigations with Drift Cards to Determine the Influence of the Wind in Surface Currents, pp. 129-139, Studies on Oceanography ed. K. Yoshida, University of Washington Press, Seattle. U ~ t J . , F. 1950. On the Theoretical Form of Ocean Swell on a Rotating Earth, Mon. Not. R. astr. Soc. Geophys. Suppl. 6, 1-8. VAN DORN, W. G. 1953. Wind Stress on an Artificial Pond. J. mar. Res. 12,219-275.