The Dispersion of Surface Contaminants by Stokes Drift In

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Acta Oceanol. Sin., 2018, Vol. 37, No. 7, P. 55–61 DOI: 10.1007/s13131-018-1243-z http://www.hyxb.org.cn E-mail: [email protected] The dispersion of surface contaminants by Stokes drift in random waves HUANG Guoxing1, 2*, LAW Wing-Keung Adrian3 1 State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China 2 State Key Laboratory of Hydrology, Water Resources and Hydraulics Engineering, Nanjing 210029, China 3 School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798, Singapore Received 8 July 2017; accepted 10 October 2017

Abstract Contaminants that are floating on the surface of the ocean are subjected to the action of random waves. In the literature, it has been asserted by researchers that the random wave action will lead to a dispersion mechanism through the induced Stokes drift, and that this dispersion mechanism may have the same order of significance comparable with the others means due to tidal currents and wind. It is investigated whether or not surface floating substances will disperse in the random wave environment due to the induced Stokes drift. An analytical derivation is first performed to obtain the drift velocity under the random waves. From the analysis, it is shown that the drift velocity is a time-independent value that does not possess any fluctuation given a specific wave energy spectrum. Thus, the random wave drift by itself should not have a dispersive effect on the surface floating substances. Experiments were then conducted with small floating objects subjected to P-M spectral waves in a laboratory wave flume, and the experimental results reinforced the conclusion drawn. Key words: dispersion, surface contaminants, random wave, Stokes drift Citation: Huang Guoxing, Law Wing-Keung Adrian. 2018. The dispersion of surface contaminants by Stokes drift in random waves. Acta Oceanologica Sinica, 37(7): 55–61, doi: 10.1007/s13131-018-1243-z

1 Introduction spectrum; ! = gk is the angular frequency, k is the wave num- Floating contaminants on the surface of the ocean, such as an ber; and z is thep vertical co-ordinate measured positively upward oil patch, are subjected to a number of dispersion mechanisms, from the mean water level. including ambient turbulence (Maksimenko, 1990; Craig and Herterich and Hasselmann (1982) suggested that this en- Banner, 1994; Huang et al., 2013) as well as shear dispersion due semble mean represents the time-average drift velocity, and that to winds, ocean currents and tidal flows (Sanderson and Pall, an individual particle at different positions under the wavy pro- 1990; Mantovanelli et al., 2012). Through the dispersive action, file will undergo an “instantaneous” drift velocity, us , such that the bulk planar size of the surface contaminants would then en- the particle will experience drift-velocity fluctuations u0s large with time. When a small size scale of up to a few kilometers (u0 = u < u > ) due to the fluctuations of the local Rayleigh- s s¡ s is considered, Herterich and Hasselmann (1982) suggested that distributed wave amplitudes in the random sea. The random the action of the random waves is a possible mechanism that has fluctuations in the drift velocity would then translate to a disper- the same order of significance comparable with the others. They sion effect following the random-walk theory, and the corres- explained that the dispersion mechanism by waves is through the ponding dispersion coefficient, D , can be obtained as random fluctuations of the wave-induced Stokes drift. Sub- sequently, most of the recent investigations on the wave-induced =< 2 > ¿; dispersion of floating contaminants on the sea surface, i.e., Mes- D u0s (2) quita et al. (1992), Giarrusso et al. (2001), Buick et al. (2001) and where ¿ represents the integral autocorrelation time of the fluctu- Pugliese Carratelli et al. (2011), were grounded in this theoretical ations. Herterich and Hasselmann (1982) further extended the development by Herterich and Hasselmann (1982). concept to a multi-directional sea, in which case the following The detailed derivation of Herterich and Hasselmann (1982) dispersion tensors were obtained. can be summarized as follows. With a random wave field in deep water, the ensemble-mean Stokes drift velocity can be expressed as 1 1 D xx D xy 6 2 2kz = ! f (!) d ! £(µ )£(µ ) < u (z) >= 2 f (k)!k e d k; 2 1 2 s (1) Ã D yx D yy ! 4g £ Z0 Z Z Z0 ¡ ¡ 2 A 1 A 2 where < us(z) > is the mean drift velocity with the angle brack- [1 + cos(µ1 µ2)] dµ1dµ2: (3) ¡ A 3 A 4 ets denoting the ensemble averaging, f (k) is the wave number µ ¶ Foundation item: The State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering Research Foundation of China under contract No. 2015491311. *Corresponding author, E-mail: [email protected]

56 HUANG Guoxing et al. Acta Oceanol. Sin., 2018, Vol. 37, No. 7, P. 55–61 where £(µ) is a frequency-independent directional spreading tial in a progressive regular wave is (e.g., Dean and Dalrymple, 2 factor; D ij a diffusion tensor; A 1 = (cos µ1 + cos µ2) ; A2 =(cos θ1 + 1991) well understood as cos µ2)(sin µ1 + sin µ2); A 3 = (cos µ1 + cos µ2)(sin µ1 + sin µ2); and 2 (1) (2) A 4 = (sin µ1 + sin µ2) . '(x; z; t) = ' (x; z; t) + ' (x; z; t) + ::: Buick et al. (2001) presented the experimental results for the a! cosh k(z + d) = sin µ + case of a pair of surface particles in a random wave basin to veri- k sinh kd · fy the theoretical predictions by Herterich and Hasselmann 3 cosh 2k(z + d) (1982). They show that the experimental results appear to agree ak sin 2µ ; (5) 8 sinh4 kd with the theoretical prediction at lower values of the particle sep- ¸ arating distance. However, the measurements deviated substan- where d is the water depth. The Eulerian velocity and displace- tially with the prediction when the distance increased. ment up to second order can then be derived as Herterich and Hasselmann (1982) theoretical development is based on the conjecture that the Stokes drift velocity fluctuates in a random manner under the multi directional random sea. In or- ' ('(1) + '(2)) u(x; z; t) = = der to address the issue, it is thus essential to review the funda- x x mental concepts related to the analysis of random waves. It is cosh k(z + d) = a! cos µ+ well known that there are two possible approaches to analyze the sinh kd · time series of a random wave field, namely the time-domain and 3 cosh 2k(z + d) ak cos 2µ ; (6) frequency-domain analysis (US Army Corps of Engineers, 2002). 4 sinh4 kd In the time-domain analysis, or commonly known also as the ¸ zero-crossing approach, the time series of the surface elevation ' ('(1) + '(2)) w(x; z; t) = = record is a composite of a number of individual waves which z z have different wave heights and periods based on the zero-cross- sinh k(z + d) = a! sin µ+ ing intervals. Longuet-Higgins (1952) shows that the probability sinh kd density of the individual zero-crossing wave height follows the · 3 sinh 2k(z + d) Rayleigh distribution. On the other hand, for the frequency-do- ak sin 2µ ; (7) 4 sinh4 kd main analysis or also commonly known as the spectral analysis, ¸ the surface elevation is expanded in the form of a Fourier series: t cosh k(z0 + d) x x 0 = u d t a sin µ0; (8) ¡ ¼ ¡ sinh kd 1 Z0 ´(t) = ai cos(kix !it + "i) i=1 ¡ t X cosh k(z0 + d) 1 z z0 = w d t a cos µ0; (9) = 2s(!i)¢!i cos(kix !it + "i); (4) ¡ ¼ sinh kd Z0 i=1 ¡ X p where µ = kx !t, and (x 0; z0) is the initial position. where a , k , ! and " are the amplitude, wave number, angular ¡ i i i i Given the Eulerian velocities and displacements, the Lag- frequency and random phase of each wave component, respect- rangian horizontal velocity can be obtained by Taylor’s expan- ively; and s(!) is the frequency spectrum. The functional form sion as implies that the time series is the superposition of an infinite number of linear waves with different frequencies. The two dif- u ferent approaches are typically used independently, although it u(x; z; t) = u(x ; z ; t) + (x x )+ 0 0 x ¡ 0 has been shown that both approaches give a similar value of a µ ¶x o; zo significant wave height in deep waters (i.e., Gōda, 2010). u (z z ) + O (ka)3 ; In Herterich and Hasselmann (1982) approach, the en- z ¡ 0 µ ¶x o; zo semble-mean Stokes drift velocity is first obtained in the fre- ³ ´ 2 cosh k(z0 + d) c(ka) quency-domain, and then the ‘instantaneous’ fluctuating drift = c (ka) cos µ0 + sinh kd 2 sinh2 kd £ velocity is evaluated in the time-domain. This cross over poses a 3 cosh 2k(z0 + d) potential ambiguity in the derived relationship for the dispersion 1 cos 2µ0+ 2 sinh2 kd ¡ effect. Here, the question of whether or not the drift velocity of · ¸ cosh 2k(z + d) the random surface waves would lead to a dispersion effect of 2 0 3 (10) c(ka) 2 + O (ka) : surface floating substances needs to be reexamined. In the fol- 2 sinh kd ³ ´ lowing, we shall show that the drift velocity obtained in the frequency domain is a time-independent value that can be con- From Eq. (10), it can also be concluded that, up to the second sidered as an instantaneous value as well, such that the random order, the Lagrangian velocity consists of a first order and second fluctuations would not exist. An experimental study will also be order oscillatory components plus a time mean Stokes drift com- presented to reinforce the conclusion. ponent of

2 Stokes drift in regular and random waves 2 cosh 2k(z0 + d) us = c(ka) : (11) 2 sinh2 kd 2.1 Stokes drift in regular waves The Stokes drift in regular waves is first derived before the The above derivations and results are well known and can be drift under the random waves is considered. The velocity poten- commonly found in the literature. We show the derivation here HUANG Guoxing et al. Acta Oceanol. Sin., 2018, Vol. 37, No. 7, P. 55–61 57

§ § in details to illustrate the fact that the drift velocity appears as a G §(! ; ! ) = g g 1 2 12 § 21 component to the instantaneous velocity, and that no period av- 1 1 = g2 k k eraging is needed to obtain the relationship in Eq. (11). Hence, ¡ 1 2 ! § ! £ · µ 1 2 ¶ the drift velocity is a time-independent quantity that can also be (1 tanh k d tanh k d)+ ¨ 1 2 considered as an instantaneous value, or in other words, it does 2 2 1 k1 k2 (15) not possess any fluctuations. Note that similar derivations can be 2 2 : 2 !1 cosh k1d § !2 cosh k2d performed for the z-direction to illustrate that such drift velocity µ ¶¸ does not exist vertically, as expected. Then

2.2 Stokes drift in random waves 2 A spectral or superposition method, which treats the surface '2 '2 2 + g = a1a2G §(!1; !2) sin(k x ! t)+ profile as a superposition of infinitude linear waves with differ- t z § ¡ § 1 2 + ent frequencies, amplitudes and phases, can be used to analyze a a1 G (!1; !1) sin 2(k1x !1t)+ 2 ¡ random wave field. Here, we shall first derive the interaction 1 a 2G +(! ; ! ) sin 2(k x ! t); (16) between two sinusoidal wave components. Subsequently, a solu- 2 2 2 2 2 ¡ 2 tion will be generalized to include the entire wave spectrum.

The surface boundary condition stipulates that the second or- where k = k1 k2; and ! = !1 !2 . § § § § der velocity potential of a progressive wave (Dean and Dalry- Consider a special solution: mple, 1991) should satisfy the following:

(2) cosh k (z + d) ' (x; z; t) = a a E § § 2'(2) '(2) '(1) 2'(1) '(1) 2'(1) 1 2 12 cosh k d £ § 2 + g = 2 + + t z ¡ x x t z z t 1 2 µ ¶ sin(k x ! t) + a1 E 1§1 1 '(1) 2'(1) 3'(1) § ¡ § 2 £ g 2 + 2 : (12) cosh 2k1(z + d) g t z z t sin 2(k1x !1t)+ µ ¶ cosh 2k1d ¡

1 2 cosh 2k2(z + d) Taking two distinct-frequency sinusoidal waves along the x- a2 E 1§2 2 cosh 2k2d £ axis, the first order velocity potential due to the superposition is sin 2(k2x !2t): (17) ¡ ga cosh k (z + d) '(1)(x; z; t) = 1 1 sin(k x ! t)+ Besides the surface condition, this special solution should ! cosh k d 1 ¡ 1 1 1 also satisfy the Laplace governing equations and the bottom ga2 cosh k2(z + d) sin(k2x !2t): (13) boundary condition: !2 cosh k2d ¡

for d < z < 0; 2Á(2) = 0; (18) Substituting Eq. (13) into Eq. (12) and on the surface, z=0, we ¡ r have Á(2) for z = d; = 0: (19) ¡ z 2'(2) '(2) 1 g2a 2k 2 + g = sin 2µ 3 1 1 (tanh2 k d 1) + t2 z 2 1 ! 1 ¡ Substituting Eq. (17) into Eqs (18) and (19), we have · 1 ¸ 2 2 2 1 g a2 k2 2 sin 2µ2 3 (tanh k2d 1) + + + 2 !2 ¡ G §(!1; !2) + G (!1; !1) + G (!2; !2) · ¸ E 1§2 = E 11 = + E 22 = + ; (20) 1 1 1 D §(!1; !2) D (!1; !1) D (!2; !2) sin(µ + µ ) 2g2a a k k + + 2 1 2 ¡ 1 2 1 2 ! ! · µ 1 2 ¶ 2 2 1 1 2 where D §(!1; !2) = gk tanh(k d) ! . 2g a a k k tanh k d tanh k d + g a a § § ¡ § 1 2 1 2 1 2 ! ! ¡ 1 2£ µ 1 2 ¶ Utilizing the same principle, if an infinite number of wave 2 2 k1 k1 components are considered, then the second order potential un- + +ga a (k ! tanh k d+k ! tanh k d) + ! ! 1 2 1 1 1 2 2 2 µ 1 2 ¶ ¸ der the random wave condition can be obtained as follow: 1 2 1 1 sin(µ1 µ2) 2g a1a2k1k2 + 2 ¡ ¡ !1 ¡ !2 · µ ¶ (2) G §(!i; !j ) cosh k (z + d) ' (x; z; t) = aiaj § 2 1 1 2g a a k k tanh k d tanh k d i j >i D §(!i; !j ) cosh k d £ 1 2 1 2 1 2 ! ¡ ! ¡ X X § µ 1 2 ¶ + 2 2 1 2 G (!i; !i) 2 k1 k1 sin(k x ! t + " ) + ai + g a a + ga a § ¡ § § 2 D (!i; !i) £ 1 2 ! ¡ ! 1 2£ i µ 1 2 ¶ X cosh 2ki(z + d) sin 2(kix !it + "i): (21) (k1!1 tanh k1d k2!2 tanh k2d) ; (14) cosh 2k d ¡ ¡ i ¸ For a random field of ocean waves, the surface displacement, where µ = k x ! t , and µ = k x ! t . velocity potential and Eulerian velocity to the second order are 1 1 ¡ 1 2 2 ¡ 2 Denote a function G where by: then given as 58 HUANG Guoxing et al. Acta Oceanol. Sin., 2018, Vol. 37, No. 7, P. 55–61

´(1)(x; t) = a cos(k x ! t + " ); aiaj (ki!i + kj !j ) i i ¡ i i (22) i 2 sinh kid sinh kj d £ X Xi Xj ¢ a g cosh k (z + d) [cos(k x 0 ! t + " ) cosh k+(z0 + d) Á(1)(x; z; t) = i i ¡ ¡ ¡ ¡ ¡ ! cosh k d £ cos(k+x 0 !+t + "+) cosh k (z0 + d)]+ i i i ¡ ¡ X (a k )2 sin(kix !it + "i); (23) i i ¡ ci 2 cos 2(kix 0 !it + "i)+ 2 sinh kid ¡ Xi G §(!i; !j ) Á(2)( ; ; ) = 2 cosh 2ki(z0 + d) 3 x z t aiaj ci(kiai) + O(ka) : D (!i; !j ) £ 2 (30) i j >i § 2 sinh kid X X Xi cosh k (z + d) § sin(k x ! t + " )+ cosh k d § ¡ § § In this expression, the first term denotes the first harmonic, § + 1 2 G (!i; !i) cosh 2ki(z + d) the third and fifth terms the second harmonic, and the second ai + 2 i D (!i; !i) cosh 2kid £ and fourth terms the interaction among the different wave com- X ponents. The last term is the drift velocity in the random wave sin 2(kix !it + "i); (24) ¡ field, which is simply the algebraic summation of the drift velo- (1) city from all the wave components, i.e., (1) Á cosh ki(z + d) u (x; z; t) = = ai!i x i sinh kid £ X 1 cos(kix !it + "i); (25) 2 cosh 2ki(z0 + d) ¡ us = ci(kiai) 2 i=1 2 sinh kid G (! ; ! ) X (2) § i j 1 u (x; z; t) = aiaj k cosh 2k(z + d) § 0 > D §(!i; !j ) £ = 2 s(!)!k d !: Xi Xj i 2 sinh2 kd (31) cosh k (z + d) Z0 § cos(k x ! t + " )+ cosh k d § ¡ § § § + In deep water condition, the surface drift at z =0 would be 2 G (!i; !i) cosh 2ki(z + d) 0 ai + i D (!i; !i) cosh 2kid £ X 1 ki cos 2(kix !it + "i); (26) ¡ us = 2 s(!)!k d !; (32) t Z0 cosh ki(z0 + d) x x 0 = udt ai ¡ ¼ i ¡ sinh kid £ which is identical to Eq. (1). Z0 X Equations (30) and (32) illustrate that when the wave energy sin(kix 0 !it + "i); (27) ¡ spectrum is defined, the corresponding drift velocity would then

t be a time-independent quantity. In other words, the derivation cosh k (z + d) implies an important outcome that the Stokes drift should not z z = !dt = a i 0 ¡ 0 » sinh k d £ possess any fluctuation component, similar to the situation un- Z i i 0 X der the regular waves. Consequently, it should not cause any dis- cos(kix 0 !it + "i); (28) ¡ persion effect to the floating substances under the random wave field, which is contrary to the suggestion by Herterich and Has- where ! = !i !j ; k = ki kj ; " = "i "j ; D §(!i; !j ) = gk selmann (1982) and others. § § § § § § § tanh(k d) ! 2 ; and § ¡ § 3 Experimental results and discussion In the above section, the derivations have shown that there 2 kikj G §(!i; !j ) = g ! (1 tanh kid tanh kj d)+ should not be a dispersion effect to the floating substances due to ¡ !i!j § £ ¨ · the surface drift in random waves. To verify this conclusion, we k 2 k 2 i j : (29) carried out an experimental study as described in the following. 2! cosh2 k d § 2! cosh2 k d µ i i j j ¶¸ 3.1 Experimental setup Again, the Lagrangian horizontal velocity can be derived from Experiments were conducted in a wave flume measuring the Taylor’s expansion: 45.00 m long, 1.55 m wide and 1.50 m height (Fig. 1). The large flume size allowed deep-water waves to be generated. One end of cosh ki(z0 + d) u(x; z; t) = ci(aiki) the flume housed a two-dimensional piston type random wave- sinh kid £ Xi maker. The wave piston was controlled directly using the DHI cos(k x ! t + " )+ i 0 ¡ i i active wave absorption control system (AWACS). Meanwhile, G §(!i; !j ) DHI wave synthesizer software was used to specify the types of aiaj k § > D §(!i; !j ) £ wave required as well as their respective parameters. At the other Xi Xj i cosh k (z0 + d) end of the wave flume, an artificial absorbent beach was built to § cosh k d £ provide an efficient dissipation of the wave energy. § Four sensitive capacitance wave probes were mounted on a cos(k x 0 ! t + " )+ § § § +¡ steel frame positioned at a range of 1.2–2.2 m from the wave 2 G (!i; !i) cosh 2ki(z0 + d) ai + paddle. The wave probes were capable of measuring wave D (!i; !i) cosh 2kid £ Xi heights to the nearest 0.5 mm. The probes had a small diameter HUANG Guoxing et al. Acta Oceanol. Sin., 2018, Vol. 37, No. 7, P. 55–61 59

2.2 m video camera 1.2 m

markers wave flume wave probes artificial beach wave maker 0.2 m ≈ 2.0 m ruler Fig. 1. Schematic layout of the wave flume (profile view). Fig. 2. Schematic layout of experimental set-up (plan view). (approximately 0.6 cm) such that their placement would not res- ult in significant modifications of the wave profile. 3.3 Experimental results Small vinyl marks with a radius of about 1 cm were used as In the experiments, the Pierson-Moscowitz (P-M) spectrum floating particles in the experiments. The vinyl marks had a dens- was used to generate the random waves. Three different tests ity of 0.9 g/cm3 and they were used to simulate the surface con- were conducted with Hs =0.03, 0.04 and 0.05 m respectively with taminant particles. They floated on the water surface without any the water depth d=0.8 m. The measured wave spectra, plotted in relative movement to the surface water particles. A ruler is fixed Fig. 3, show that the spectrum generated in the tank agrees well on the wall of one side of the wave flume to determine the posi- with the theoretical P-M spectrum. In order to verify the repeti- tion of the markers. On the opposite side, a digital video camera tion of experiments, each of the three tests were repeated three mounted on a tripod was positioned beside the measurement times, and similar results were obtained. area to capture the images of the particles’ movement. The field The distances ΔL between the marker particles determined of view of the camera covered a span of about 2 m, in which the by the coordinate-difference in the video recordings at different position of the small vinyl marks and their coordinates in the im- time is shown in Fig. 4. It can be observed that they only oscillate ages can be identified. The schematic layout of the experimental within a small range (±0.05 m) in these experiments during the set-up is shown in Fig. 2. test duration. These experimental observations are distinctly different from those of Buick et al. (2001) who reported that the rel- 3.2 Experimental procedures ative distance of two marker particles increases with time. In Two vinyl marks were placed on the water surface along the their measurements, Buick et al. (2001) state that the turbulent wave flume. According to Eq. (27), the first order horizontal dis- diffusion have a significant effect, which implies that the wave in placement should be of the same order as the wave height (max- their experiments does not match potential flow anymore. imum Hs=0.05 m in this study). The initial distance for the two However, the wave generated in the current study meets the lin- marks was thus set to be 0.2 m to avoid their collision. An experi- ear wave criteria. This can be the reason leading to the difference. ment was initiated only after the water was sufficiently still. This The comparison between the experimental observations and can be judged by visually inspecting the motion of the marks for Herterich and Hasselmann’s (1982) predictions is also presented 1 min or 2 min. Any discernible movement would indicate the in Fig. 4. Following Herterich and Hasselmann’s (1982) theory, presence of a residual current. After it was ensured that there after 100 s, the relative distance between the two markers would were no residual currents, the digital video camera was switched have increased by 8, 10 and 12 cm corresponding to the signific- on and the wave generator was activated. ant wave height 0.03, 0.04 and 0.05 m, respectively (see Appendix The experimental duration was chosen to be less than 100 s to A). Clearly, this predicted increase was not observed in our ex- avoid the effects by the reflected waves. Prior to the study, we had periments. The experimental results thus reinforce the analysis examined the effectiveness of the artificial beach installed in that the stochastic Stokes drift in the random waves does not lead terms of dissipating the incoming waves. It was found that the to the surface dispersion of floating substances. beach was quite effective and the reflected wave height was less than 10% within the range of the wave conditions tested. Given 4 Conclusions the wave period of 1.0 s, the group velocity of the wave train The derivations of the Lagrangian velocities in both regular would be equal to 0.78 m/s. The distance between the test span and random waves performed in this study illustrate that the and the end of the flume was approximately 30 m. Hence, it Stokes drift velocity is a time-independent quantity that is among would take about 80 s for the wave train to cover the distance to- the components of the Lagrangian velocities, and thus it can also and-fro. With the experimental duration less than 100 s, the re- be considered as an instantaneous value. In this regards, the drift flected waves were thus not expected to have a significant effect. velocity should not possess any fluctuation once a wave spec-

0.8 1.6 3.0 spectrum produced by piston spectrum produced by piston spectrum produced by piston 0.7 a 1.4 b c P-M spectrum (theory) P-M spectrum (theory) 2.5 P-M spectrum (theory) s s s 1.2 · · 0.6 · 2 2 2 2.0 m m 0.5 m 1.0 4 4 4 − − − 0.4 0.8 1.5 /10 /10 /10 0.6 0.3 1.0 s(f) s(f) s(f) 0.2 0.4 0.1 0.2 0.5 0 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 0 1 2 3 4 5 f/Hz f/Hz f/Hz

Fig. 3. Measured wave spectrum (P-M). a. Hs=0.03 m, b. Hs=0.04 m and c. Hs=0.05 m. 60 HUANG Guoxing et al. Acta Oceanol. Sin., 2018, Vol. 37, No. 7, P. 55–61

0.20 References experimental results 0.15 a Buick J M, Morrison I G, Durrani T S, et al. 2001. Particle diffusion on H and H 0.10 a three-dimensional random sea. Experiments in Fluids, 30(1): 0.05 88–92

/m 0.00 L Craig P D, Banner M L. 1994. Modeling wave-enhanced turbulence in ∆ -0.05 0 10 20 30 40 50 60 70 80 90 100 the ocean surface layer. Journal of Physical Oceanography, -0.10 t/s 24(12): 2546–2559 -0.15 Dean R G, Dalrymple R A. 1991. Water Wave Mechanics for Engin- -0.20 eers and Scientists. Singapore: World Scientific Publishing 0.20 Giarrusso C C, Carratelli Pugliese E, Spulsi G. 2001. On the effects of experimental results b 0.15 wave drift on the dispersion of floating pollutants. Ocean En- H and H 0.10 gineering, 28(10): 1339–1348 0.05 Gōda Y. 2010. Random Seas and Design of Maritime Structures. 3rd

/m 0.00

L ed. New Jersey: World Scientific Publishing Company, 416–479 ∆ -0.05 0 10 20 30 40 50 60 70 80 90 100 Herterich K, Hasselmann K. 1982. The horizontal diffusion of tracers -0.10 t/s by surface waves. Journal of Physical Oceanography, 12(7): -0.15 704–711 -0.20 Huang Chuanjian, Qiao Fangli, Wei Zexun. 2013. Effects of the sur- 0.20 face wave-induced mixing on circulation in an isopycnal-co- experimental results c ordinate oceanic circulation model. Acta Oceanologica Sinica, 0.15 H and H 0.10 32(5): 7–14 0.05 Longuet-Higgins M S. 1952. On the statistical distribution of the

/m height of sea waves. Journal of Marine Research, 11(3): 245–266

L 0.00 70 ∆ -0.05 0 10 20 30 40 50 60 80 90 100 Maksimenko N A. 1990. Comparative analysis of Lagrangian statistic- -0.10 t/s al characteristics for synoptic-scale currents in hydrophysical -0.15 study areas. Oceanology, 30(1): 5–9 -0.20 Mantovanelli A, Heron M L, Heron S F, et al. 2012. Relative dispersion of surface drifters in a barrier reef region. Journal of Geo- Fig. 4. The comparison of the changes in the relative distance physical Research, 117(C11): C11016 the comparison of the results between the two markers, ΔL, Mesquita O N, Kane S, Gollub J P. 1992. Transport by capillary waves: between the experimental measurements and Herterich and fluctuating stokes drift. Physical Review A, 45(6): 3700–3705 Pugliese Carratelli E, Dentale F, Reale F. 2011. On the effects of wave- Hasselmann’s (1982) theory. a. Hs=0.03 m, b. Hs=0.04 m and c. induced drift and dispersion in the deepwater horizon oil spill. Hs=0.05 m. In: Liu Yonggang, Macfadyen A, Ji Zhengang, et al., eds. Monit- oring and Modeling the Deepwater Horizon Oil Spill: A Record- trum is defined. Since the fluctuation is a necessary mechanism Breaking Enterprise. Geophysical Monograph Series. Washing- for the dispersion effect predicted by Herterich and Hasselmann ton, DC: American Geophysical Union, 197–204 (1982), the existence of the Stokes drift in the random waves Sanderson B G, Pal B K. 1990. Patch diffusion computed from Lag- rangian data, with application to the Atlantic equatorial under- should therefore not be able to induce a dispersion effect on sur- current. Atmosphere-Ocean, 28(4): 444–465 face floating substances. The experimental findings also verify US Army Corps of Engineers. 2002. Coastal Engineering Manual. this conclusion. USACE Publications, Philadelphia, US HUANG Guoxing et al. Acta Oceanol. Sin., 2018, Vol. 37, No. 7, P. 55–61 61

Appendix: The surface dispersion coefficient under the random waves with the P-M spectrum using Herterich and Hasselmann (1982) theory The dispersion coefficient in the 3-D random wave condition formulated by Herterich and Hasselmann (1982) has been shown in

Eq. (3). When the waves only travel in one direction such as in a wave flume, the spreading functions G(θ1) and G(θ2) will become δ- function:

µ1 = 0; G(µ1) = 1 G(µ1) d µ1 = 1; 0 µ = 0; (A1) 1 6 Z n ¡

µ2 = 0; G(µ2) = 1 G(µ2) d µ2 = 1: 0 µ = 0; (A2) 2 6 Z n ¡

Substituting Eqs (A1) and (A2) into the wave direction terms in Eq. (3), the dispersion coefficient on 2-D condition can then be obtained as

4 1 D = !6s 2(!) d !: xx g2 (A3) Z0

0:78 3:11 Substituting the P-M spectrum, i.e., s(!) = exp into the above equation: !5 ¡ !4H 2 µ s ¶

1 4 2 4 6:22 D = 0:78 !¡ exp d!: xx 2 4 2 (A4) g ¡ ! H s Z0 µ ¶

Given the value of the significant wave height Hs, the dispersion coefficient can then be solved numerically by Eq. (A4). The change of the relative distance between the two markers, ΔL with time t can then be computed with the significant wave height for the three tests.