Taylor Dispersion of Contaminants by Dual-peak Spectral Random Waves

Recent extensive and important studies have provided detailed information and compelling evidence on how the presence of waves influences the vertical diffusivity/dispersivity in the coastal environment, which can affect various water quality considerations such as the distribution of suspended sediments in the water column as well as the potential of eutrophication. Comparatively, how the presence of waves influences the horizontal diffusivity/dispersivity has received only scant attention in the literature. Our previous works investigated the role played by the Taylor mechanism due to the wave-induced drift profile which leads to the longitudinal dispersion of contaminants in the horizontal direction, under regular sinusoidal waves and random waves with single-peak spectra. Natural waves in the coastal environment, however, often possess dual-peak spectra, comprising both higher frequency wind waves and lower frequency swells. In this study, the Taylor dispersion of contaminants under random waves with dual-peak spectra is examined through analytical derivation and numerical calculations. The effects of various dual-peak spectral parameters on the horizontal dispersion, including the proportion of lower frequency energy, peak frequency ratio and spectral shape parameter, are investigated. The results show that the relative energy distribution between the dual peaks has the most significant effect. Compared with single-peak spectra with equivalent energy, the Taylor dispersion with dual-peak spectra is stronger when the lower frequency is close to the peak frequency of the single-peak spectrum, and weaker with the higher frequency instead. Thus, it can be concluded that with a dual-peak wave spectrum, wind-dominated seas with higher frequency lead to stronger dispersion in the horizontal direction than swell-dominated seas with lower frequency.


Introduction
The water quality in the nearshore coastal environment can have a direct impact on human health and is thus a matter of significant concern. Accurate assessment and predictions on the mass transport of pollutants in the coastal waters are important for the design of coastal engineering facilities towards the environmental protection. The main mechanisms for the environmental transport analysis include the advection, dispersion and reaction processes involved.
One of the key factors that affect the advection and dispersion in the coastal environment is the presence of waves, including wind waves and swells. Recent extensive and important studies (e.g. Qiao et al., 2016;Dai et al., 2010) have provided detailed information and compelling evidence of how the presence of waves influences the vertical dispersion in the coastal environment, which can affect various water quality considerations such as the distribution of suspended sediments in the water column as well as the potential of eutrophication (Wong et al., 2007). Comparatively, how the presence of waves influences the horizontal dispersion has received only scant attention in the literature so far. In particular, it is anticipated that if the wave duration is sufficiently long, the horizontal dispersion by waves can potentially be dominated by the longitudinal dispersion due to the Taylor mechanism as discussed in the following.
For uni-direction flows, there have been extensive researches on the Taylor dispersion of contaminants since 1953 when Taylor (1953) derived the longitudinal disper-sion coefficient for soluble contaminants flowing through a tube. He then verified this prediction with experiments of turbulent flows through long straight pipes. Fischer et al. (1979) later on extended the Taylor's theory to the longitudinal dispersion in rivers along the flow direction, and obtained a series of formula on the dispersion coefficient with respect to the river flow characteristics. The analysis of dispersion in the coastal environment with oscillating tidal flows also began in 1953 (Longuet-Higgins, 1953), and subsequently the analysis was expanded from one dimension to three dimensions, single factor to multi-factor influence, and also simple system to multi-system coupling (e.g. Madsen, 1978;Smith, 1983;Yasuda, 1984). Later, the Taylor dispersion principle was applied to oscillatory shear currents (e.g. van den Broeck, 1990;Aris, 1956;Chikwendu, 1986). Subsequently, Law (2000) adopted the approach of Van Den Broeck in a stochastic manner with the Fokker-Plank formulation to derive the theoretical formula of horizontal dispersion coefficient under regular surface waves, including the Taylor effect with the vertical profile of wave-induced drift along the water column. The derivation incorporated various drift profiles, including Stokes drift, profile with the viscous effect due to water viscosity, and profile with a contaminated water surface. The predictions were verified with the rigid-lid assumptions using the random walk numerical approach. Subsequently, a follow-up study by Zhang et al. (2003) found that the dispersion coefficient would increase by relaxing the rigid-lid assumptions due to the wavy water surface. In recent years, Huang and Law (2011) analyzed the Taylor dispersion under more realistic random waves in the costal environment, and beyond the idealized configuration of regular sinusoidal waves. Their analysis of the horizontal dispersion coefficient due to random waves included both developed and developing sea states. An important conclusion from their study is that the longitudinal dispersion by a random sea state in the horizontal direction can be significantly stronger (by over an order of magnitude) than an equivalent regular sea state with the same wave energy density.
The above results by Huang and Law (2011) implied that it is critical to incorporate the sea state parameters to analyze how the presence of waves affects the horizontal dispersion in coastal waters. The limited existing works reviewed above considered only regular waves or single-peak wave spectra. In the real coastal environment, many in-situ measurements had shown that the natural sea state often possesses a dual-peak spectrum (e.g. Huang and Hu, 1988 in Kiaochow Bay; Fan et al., 1992 in Western Pacific Ocean;Feng et al., 2002 in Shanghai coastal region; Ren et al., 2014 in South China Sea;Semedo et al., 2015 in Nordic Sea andFontaine et al., 2013 off the sea of West Africa, and many more). The dual peaks in the spectrum can be rather distinct, and are mostly due to the presence of both wind waves and swells. There is thus a gap towards this realistic setting.
The objective of the present study is therefore to analyze the horizontal dispersion due to the presence of random waves with dual-peak spectra addressing in particular the Taylor mechanism. We follow the approach developed in Huang and Law (2011) for random waves with singlepeak spectra, and extend the analysis towards dual-peak spectra. The six-parameter representation of the wave energy spectrum by Ochi and Hubble (1976) is adopted for its flexibility. The Ochi and Hubble spectrum consists of two parts for the lower and higher frequency components, respectively. Each component is expressed in terms of three parameters, and the total spectrum is taken as a linear combination of the two. Thus, the resulting expression for the dual-peak spectral shape can represent the combination of the lower frequency swells with the higher frequency windgenerated waves. The spectral shape can also address almost all development stages of the sea state in a storm.
In the following, a dimensionless representation of the six-parameter spectrum is first derived. The longitudinal dispersion in the horizontal direction due to this spectrum is then analyzed with MATLAB. The dual-peak predictions are first verified with the asymptotic single-peak predictions by Huang and Law (2011). Subsequently, the effect of various dual-peak spectral parameters on the horizontal dispersion is examined, including the proportion of lower frequency energy, peak frequency ratio and spectral shape parameter.

Analysis
The Ochi and Hubble wave spectrum s(ω) was modeled as a sum of two Γ-distributions: (1) where ω is the wave angular frequency; Γ(λ) is the gamma function; H s is the significant wave height; ω m is the peak angular frequency; and λ is the spectral shape parameter that controls the extent which the spectral energy is concentrated around the peak frequency. j=1, 2 represents the lower and higher frequency components, respectively.
Based on the Rice's theory (Yu, 2000), the significant wave height and wave energy m 0 of the dual-peak spectrum can be obtained as: (2) Thus, the spectrum can be normalized by the higher frequency ω m2 as: s(ω)ω where is the dimensionless dual-peak spectrum and is the dimensionless frequency.
By substituting Eqs. (1) and (4) into Eq. (3), the dimensionless spectrum can thus be represented as: Parameter p can be defined to replace the fourth power of the ratio of lower peak-frequency ω m1 and higher peakfrequency ω m2 on the RHS of Eq. (5) as follows: We shall now extend the analysis of the dual peak spectral shape based on the results of Huang and Law (2011), which derived the horizontal dispersion under single-peak random waves as: where K is Taylor dispersion coefficient; D is the background ambient diffusivity/dispersivity; ε m is the water depth factor; g is the gravitational acceleration; Ω r is the wave-dispersion parameter for random waves; T m is the peak period (= 2π/ω m ); and d is the water depth. The water depth factors of f 1 (ε m ) and f 2 (ε m ) are also derived as: where and z'= z/d, z being the vertical coordinate.ω

S(ω)
With the representation of the dual-peak spectrum in Eq. (1), the governing Eq. (7) remains the same for the individual components. Given Eqs. (10) and (11) above and that and can be expressed by Eqs. (4b) and (5), respectively for dual-peak spectrum, we can make use of the same derivation by replacing ε m and Ω r in the governing Eq. (7) with Eqs. (12) and (13) as follows: Subsequently, by using MATLAB, the longitudinal dispersion coefficient can then be computed with various spectral parameters for the dual-peak spectrum, and the assessment of their effects can then be made.

Validation
To validate the above dual-peak predictions, an asymptotic case is first considered by taking either the lower or higher frequency wave energy to be zero, i.e. λ 1 = 1, λ 2 = 0 or λ 1 = 0, λ 2 = 1. With lower-frequency energy λ 1 = 0, the dualpeak spectrum is then identical to the P-M spectrum (Yu, 2000). As one of the most representative energy spectra, P-M spectrum was proposed for a wind generated fully developed sea state based on the similarity theory of Kitaigorodskii and site measurements, where the peak frequency ω m and significant wave height H s have a coupled relationship as: (14) Based on the MATLAB calculations, the variation of f(ε m ) versus ε m and K/D versus Ω r is plotted in Figs. 1 and 2,   Fig. 1. Variation of f(ε m ) versus ε m (wind wave spectrum/P-M spectrum). HUANG Guo-xing et al. China Ocean Eng., 2019, Vol. 33, No. 5, P. 537-543 539 respectively. It can be seen that the asymptotic results are identical to those reported by Huang and Law (2011) for the P-M spectrum (Fig. 3). Thus, the results validate the model for the dual-peak spectrum.

Results
With the validation, the effects of various dual-peak spectral parameters on the magnitude of the longitudinal dispersive coefficient are now discussed in this section. The parameters include the proportion of lower frequency energy, peak frequency ratios and spectral shape parameter.

Effect of proportion of low frequency spectrum energy
We shall assume the example values of ω m1 = 0.6 (i.e. wave period T = 10.5 s) and ω m2 = 1.2 (i.e. T = 5.25 s) for the evaluation. The values of significant wave height are also shown in Table 1. Correspondingly, the proportion of lower frequency spectrum energy is 0, 20%, 50%, 80% and 1, respectively. Fig. 4 shows the dimensionless dual-peak spectra with various proportions of lower frequency wave energy above. The predictions based on the numerical analysis by MAT-LAB are shown in Figs. 5 and 6. From the figures, the dispersion coefficient with dual-peak spectra is always smaller than the purely wind sea state with the equivalent wave energy, but larger than the purely swell sea state. This can be attributed directly to the fact that the longitudinal dispersion due to the Taylor mechanism increases generally with higher wave frequency (Law, 2000). Thus, it can be expected that for dual-peak spectra, the Taylor dispersion would be stronger with a larger proportion of the wave energy in the higher frequency component.     4.2 Effect of peak frequency ratio To analyze the effect of peak frequency ratios, we shall assume the example values of lower frequency energy ratio of 20% and 80% for illustration. The corresponding values of significant wave height and peak frequency ratios are shown in Table 2. Fig. 7 shows the dimensionless dual-peak spectra with various ω m1 /ω m2 for lower frequency energy of 20% and 80%. In the simulations, the peak frequency ratios can be controlled by fixing the peak frequency of wind-sea and adjusting the peak frequency of swell. The results show that the dispersion coefficient increases with the peak frequency of swell. Again, this can be attributed to the fact that Taylor dispersion due to wave motion is generally stronger with higher frequency.
The same trend can be observed in Figs. 10 and 11 for the case with the lower frequency energy of 80%. By comparing Fig. 8 with Fig. 9, it can be observed directly that the higher frequency component contributes stronger dispersive effect, which reinforces the previous results.

Effect of spectral shape parameter
To evaluate the effect of the spectral shape parameter, we have chosen the following example values for illustration: ω m2 = 1.2 (T = 5.25 s) and ω m1 = 0.6 (T = 10.5 s), H s1 = H s2 =0.8 m (i.e. the same spectral energy for the dual com- Fig. 10. Variation of f(ε m ) versus ε m for spectra with various peak frequency ratios (lower frequency energy of 80%). Fig. 11. Variation of K/D versus Ω r for spectra with various peak frequency ratios (lower frequency energy of 80%).   8. Variation of f(ε m ) versus ε m for spectra with various peak frequency ratios (lower frequency energy of 20%). Fig. 9. Variation of K/D versus Ω r for spectra with various peak frequency ratios (lower frequency energy of 20%). ponents), and ε m = 2.0. Both λ 1 and λ 2 are set as 1, 3.3 and 7 for the evaluation (Table 3).
Figs. 12a and 12b present the dimensionless dual-peak spectra with varying λ 1 and λ 2 , respectively. Figs. 13 and 14 show the variation of f(ε m ) and the resulting dispersion coefficient K, respectively, for various λ 1 . The results show that the dispersion coefficient increases monotonically with λ 1 , indicating that the longitudinal dispersive effect is more prominent with larger spectral shape parameters.
The same trend can be observed in Figs. 15 and 16 for the case with varying λ 2 . As comparing Fig. 13 with Fig. 15, the variation of λ 2 affects the dispersion coefficient more significantly than the variation of λ 1 . This reinforces the conclusion that the higher frequency component contributes more significantly to the longitudinal dispersion.

Effect of water depth factor
To evaluate the effect of the water depth factor ε m , we have chosen the example values of ω m2 = 1.2, ω m1 = 0.6, H s1 = 0.5 m, H s2 = 1.0 m, λ 1 = λ 2 = 1. With ε m of 0.4, 0.8, 1.2, 2.0, 2.8, the relationship between K/D and Ω r is plotted in Fig. 17. It can be observed that the longitudinal dispersion increases with ε m significantly. Also, the dispersion coefficient is negligible compared with the ambient dispersion with Ω r < 2, but becomes significantly stronger beyond this range. Comparing the current results with Huang and Law (2011), the magnitude of the longitudinal dispersion   13. Variation of f(ε m ) versus ε m for spectra with various shape parameters (fixed λ 2 , varying λ 1 ).

Fig. 14.
Variation of K/D versus Ω r for spectra with various shape parameters (fixed λ 2 , varying λ 1 ). Fig. 15. Variation of f(ε m ) versus ε m for spectra with various shape parameters (fixed λ 1 , varying λ 2 ). Fig. 16. Variation of K/D versus Ω r for spectra with various shape parameters (fixed λ 1 , varying λ 2 ). coefficient in dual-peak random waves is generally lower than that in the wind wave state (P-M spectrum) with the same magnitude of wave energy. Hence, it can be inferred that the longitudinal disperse by the higher frequency wind waves is more significant than swells.

Conclusion
In the present study, the horizontal dispersion in coastal waters due to the presence of random waves with dual-peak spectra is investigated (with parameterization following the Ochi-Hubble spectrum). The results show that the Taylor dispersion coefficient generally increases with the water depth factor ε m . It also increases with the peak frequency ratio ω m1 /ω m2 although the effect is relatively minor, with the higher frequency component shape parameter λ 2 having a stronger influence.
Overall, the variation of the Taylor dispersion by dualpeak spectra shows the same tendency to that of wind sea spectra with the same peak frequency of the higher frequency, but the magnitude is smaller if the energy proportion is lower. Compared with single-peak spectra with equivalent energy, the Taylor dispersion with dual-peak spectra is stronger when the lower frequency is close to the peak frequency of the single-peak spectrum, and weaker with the higher frequency instead. Overall, it can be concluded that with a dual-peak wave spectrum, wind-dominated seas with higher frequency lead to stronger dispersion in the horizontal direction than swell-dominated seas with lower frequency.