Home Journals Articles Updates About
CN
image
收藏切换
The dispersion of surface contaminants by Stokes drift in random waves
收藏切换
PDF
Guoxing HUANG1, 2, *, Wing-Keung Adrian LAW3
Acta Oceanologica Sinica | 2018, 37(7) : 55 - 61
Less
收藏切换
Acta Oceanologica Sinica | 2018, 37(7): 55-61
• Ocean Engineering •
The dispersion of surface contaminants by Stokes drift in random waves
Full
Guoxing HUANG1, 2, *, Wing-Keung Adrian LAW3
Affiliations
  • 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
Published: 2018-07-25 doi: 10.1007/s13131-018-1243-z
Outline
收藏切换

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.

dispersion  /  surface contaminants  /  random wave  /  Stokes drift
Guoxing HUANG, Wing-Keung Adrian LAW. The dispersion of surface contaminants by Stokes drift in random waves[J]. Acta Oceanologica Sinica, 2018 , 37 (7) : 55 -61 . DOI: 10.1007/s13131-018-1243-z
Full Text
Less
1 Introduction
Less
Floating contaminants on the surface of the ocean, such as an oil patch, are subjected to a number of dispersion mechanisms, including ambient turbulence (Maksimenko, 1990; Craig and Banner, 1994; Huang et al., 2013) as well as shear dispersion due to winds, ocean currents and tidal flows (Sanderson and Pall, 1990; Mantovanelli et al., 2012). Through the dispersive action, the bulk planar size of the surface contaminants would then enlarge with time. When a small size scale of up to a few kilometers is considered, Herterich and Hasselmann (1982) suggested that the action of the random waves is a possible mechanism that has the same order of significance comparable with the others. They explained that the dispersion mechanism by waves is through the random fluctuations of the wave-induced Stokes drift. Subsequently, most of the recent investigations on the wave-induced dispersion of floating contaminants on the sea surface, i.e., Mesquita et al. (1992), Giarrusso et al. (2001), Buick et al. (2001) and Pugliese Carratelli et al. (2011), were grounded in this theoretical development by Herterich and Hasselmann (1982).
The detailed derivation of Herterich and Hasselmann (1982) can be summarized as follows. With a random wave field in deep water, the ensemble-mean Stokes drift velocity can be expressed as
$ < {{u_{\mathop{\rm s}\nolimits} }(z)} > = 2\int\limits_0^\infty {f(k)\omega k{{\mathop{\rm e}\nolimits} ^{2kz}}{\mathop{\rm d}\nolimits} k} ,$
where $ < {u_{\rm s}}(z) > $ is the mean drift velocity with the angle brackets denoting the ensemble averaging, f (k) is the wave number spectrum; $\omega = \sqrt {gk} $is the angular frequency, $k$ is the wave number; and $z$ is the vertical co-ordinate measured positively upward from the mean water level.
Herterich and Hasselmann (1982) suggested that this ensemble mean represents the time-average drift velocity, and that an individual particle at different positions under the wavy profile will undergo an “instantaneous” drift velocity, ${u_{\mathop{\rm s}\nolimits} }$, such that the particle will experience drift-velocity fluctuations ${u'_{\rm s}}$ (${u'_{\rm s}} = {u_{\rm s}} - < {u_{\rm s}} > $) due to the fluctuations of the local Rayleigh-distributed wave amplitudes in the random sea. The random fluctuations in the drift velocity would then translate to a dispersion effect following the random-walk theory, and the corresponding dispersion coefficient, $D$, can be obtained as
$D = <{u'_{\text{s}}}^2 > \tau ,$
where $\tau $ represents the integral autocorrelation time of the fluctuations. Herterich and Hasselmann (1982) further extended the concept to a multi-directional sea, in which case the following dispersion tensors were obtained.
$\begin{gathered} \left( {\begin{array}{*{20}{c}} {{D_{xx}}} \\ {{D_{yx}}} \end{array}\quad \begin{array}{*{20}{c}} {{D_{xy}}} \\ {{D_{yy}}} \end{array}} \right){\text{ = }}\frac{\text{π} }{{4{g^2}}}\int\limits_0^\infty {{\omega ^6}{f^2}(\omega )\operatorname{d} \omega \int\limits_{ - \text{π} }^\text{π} {\int\limits_{ - \text{π} }^\text{π} {\varTheta ({\theta _1})\varTheta ({\theta _2})} } }\times \\ \qquad\qquad\qquad \quad \,\, {\left[ {1 + \cos ({\theta _1} - {\theta _2})} \right]^2} \left( {\begin{array}{*{20}{c}} {{A_1}}& {{A_2}} \\ {{A_3}}& {{A_4}} \end{array}} \right){\text{d}}{\theta _1}{\text{d}}{\theta _2} .\end{gathered} $
where $\varTheta (\theta)$ is a frequency-independent directional spreading factor; ${D_{ij}}$ a diffusion tensor; ${A_1} = {(\cos {\theta _1} + \cos {\theta _2})^2}$; A2 =(cos θ1 +$\cos {\theta _2})(\sin {\theta _1} + \sin {\theta _2}) $; ${A_3} = (\cos {\theta _1} + \cos {\theta _2})(\sin {\theta _1} + \sin {\theta _2}) $; and ${A_4} = {(\sin {\theta _1} + \sin {\theta _2})^2}$.
Buick et al. (2001) presented the experimental results for the case of a pair of surface particles in a random wave basin to verify the theoretical predictions by Herterich and Hasselmann (1982). They show that the experimental results appear to agree with the theoretical prediction at lower values of the particle separating distance. However, the measurements deviated substantially with the prediction when the distance increased.
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 order to address the issue, it is thus essential to review the fundamental concepts related to the analysis of random waves. It is well known that there are two possible approaches to analyze the time series of a random wave field, namely the time-domain and frequency-domain analysis (US Army Corps of Engineers, 2002). In the time-domain analysis, or commonly known also as the zero-crossing approach, the time series of the surface elevation record is a composite of a number of individual waves which have different wave heights and periods based on the zero-crossing intervals. Longuet-Higgins (1952) shows that the probability density of the individual zero-crossing wave height follows the Rayleigh distribution. On the other hand, for the frequency-domain analysis or also commonly known as the spectral analysis, the surface elevation is expanded in the form of a Fourier series:
$\begin{split} & \eta (t) = \sum\limits_{i = 1}^\infty {{a_i}\cos ({k_i}x - {\omega _i}t + {\varepsilon _i})} \\ &\qquad = \sum\limits_{i = 1}^\infty {\sqrt {2s({\omega _i})\Delta {\omega _i}} \cos ({k_i}x - {\omega _i}t + {\varepsilon _i})},\end{split} $
where ${a_i}$, ${k_i}$, ${\omega _i}$ and ${\varepsilon _i}$ are the amplitude, wave number, angular frequency and random phase of each wave component, respectively; and $s(\omega)$ is the frequency spectrum. The functional form implies that the time series is the superposition of an infinite number of linear waves with different frequencies. The two different approaches are typically used independently, although it has been shown that both approaches give a similar value of a significant wave height in deep waters (i.e., Gōda, 2010).
In Herterich and Hasselmann (1982) approach, the ensemble-mean Stokes drift velocity is first obtained in the frequency-domain, and then the ‘instantaneous’ fluctuating drift velocity is evaluated in the time-domain. This cross over poses a potential ambiguity in the derived relationship for the dispersion effect. Here, the question of whether or not the drift velocity of the random surface waves would lead to a dispersion effect of surface floating substances needs to be reexamined. In the following, we shall show that the drift velocity obtained in the frequency domain is a time-independent value that can be considered as an instantaneous value as well, such that the random fluctuations would not exist. An experimental study will also be presented to reinforce the conclusion.
2 Stokes drift in regular and random waves
Less
2.1 Stokes drift in regular waves
The Stokes drift in regular waves is first derived before the drift under the random waves is considered. The velocity potential in a progressive regular wave is (e.g., Dean and Dalrymple, 1991) well understood as
$\begin{split} \varphi (x, z, t) = &{\varphi ^{(1)}}(x, z, t) + {\varphi ^{(2)}}(x, z, t) +... \\ = & \frac{{a\omega }}{k}\left[\frac{{\cosh k(z + d)}}{{\sinh kd}}\sin \theta \right. + \\&\left. \frac{3}{8}ak\frac{{\cosh 2k(z + d)}}{{{{\sinh }^4}kd}}\sin 2\theta \right], \\ \end{split} $
where d is the water depth. The Eulerian velocity and displacement up to second order can then be derived as
$\begin{split} u(x, z, t) = & \frac{{\text{∂} \varphi }}{{\text{∂} x}} = \frac{{\text{∂} ({\varphi ^{(1)}} + {\varphi ^{(2)}})}}{{\text{∂} x}} \\ = & a\omega \left[\frac{{\cosh k(z + d)}}{{\sinh kd}}\cos \theta + \right. \\ & \left.\frac{3}{4}ak\frac{{\cosh 2k(z + d)}}{{{{\sinh }^4}kd}}\cos 2\theta \right] , \\ \end{split} $
$\begin{split} w(x, z, t) = & \frac{{\text{∂} \varphi }}{{\text{∂} z}} = \frac{{\text{∂} ({\varphi ^{(1)}} + {\varphi ^{(2)}})}}{{\text{∂} z}} \\ = & a\omega \left[\frac{{\sinh k(z + d)}}{{\sinh kd}}\sin \theta + \right. \\& \left. \frac{3}{4}ak\frac{{\sinh 2k(z + d)}}{{{{\sinh }^4}kd}}\sin 2\theta \right], \end{split} $
$x - {x_0} = \int\limits_0^t {u{\mathop{\rm d}\nolimits} t} \approx - a\frac{{\cosh k({z_0} + d)}}{{\sinh kd}}\sin {\theta _0}, $
$z - {z_0} = \int\limits_0^t {w{\mathop{\rm d}\nolimits} t} \approx a\frac{{\cosh k({z_0} + d)}}{{\sinh kd}}\cos {\theta _0}, $
where $\theta = kx - \omega t$, and $({x_0}, {z_0})$ is the initial position.
Given the Eulerian velocities and displacements, the Lagrangian horizontal velocity can be obtained by Taylor’s expansion as
$\begin{split} u(x, z, t) = & u({x_0}, {z_0}, t) + {\left({\frac{{\text{∂} u}}{{\text{∂} x}}} \right)_{{x_o}, \,\, {z_o}}}(x - {x_0}) + \\& {\left({\frac{{\text{∂} u}}{{\text{∂} z}}} \right)_{{x_o}, \,\, {z_o}}}(z - {z_0}) + O\left({{{(ka)}^3}} \right); \\ = & c\, \left({ka} \right)\frac{{\cosh k({z_0} + d)}}{{\sinh kd}}\cos {\theta _0} + \frac{{c{{(ka)}^2}}}{{2{{\sinh }^2}kd}} \times \\& \left[\frac{{3\cosh 2k({z_0} + d)}}{{2{{\sinh }^2}kd}} - 1 \right]\cos 2{\theta _0} + \\& c{(ka)^2}\frac{{\cosh 2k({z_0} + d)}}{{2{{\sinh }^2}kd}} + O\left({{{(ka)}^3}} \right).\end{split} $
From Eq. (10), it can also be concluded that, up to the second order, the Lagrangian velocity consists of a first order and second order oscillatory components plus a time mean Stokes drift component of
${u_{\mathop{\rm s}\nolimits} } = c{(ka)^2}\frac{{\cosh 2k({z_0} + d)}}{{2{{\sinh }^2}kd}}.$
The above derivations and results are well known and can be commonly found in the literature. We show the derivation here in details to illustrate the fact that the drift velocity appears as a component to the instantaneous velocity, and that no period averaging is needed to obtain the relationship in Eq. (11). Hence, the drift velocity is a time-independent quantity that can also be considered as an instantaneous value, or in other words, it does not possess any fluctuations. Note that similar derivations can be performed for the z-direction to illustrate that such drift velocity does not exist vertically, as expected.
2.2 Stokes drift in random waves
A spectral or superposition method, which treats the surface profile as a superposition of infinitude linear waves with different frequencies, amplitudes and phases, can be used to analyze a random wave field. Here, we shall first derive the interaction between two sinusoidal wave components. Subsequently, a solution will be generalized to include the entire wave spectrum.
The surface boundary condition stipulates that the second order velocity potential of a progressive wave (Dean and Dalrymple, 1991) should satisfy the following:
$\begin{split}& \frac{{{\text{∂} ^2}{\varphi ^{(2)}}}}{{\text{∂} {t^2}}} + g\frac{{\text{∂} {\varphi ^{(2)}}}}{{\text{∂} z}} = - 2\left({\frac{{\text{∂} {\varphi ^{(1)}}}}{{\text{∂} x}}\frac{{{\text{∂} ^2}{\varphi ^{(1)}}}}{{\text{∂} x\text{∂} t}}} \right. + \left. {\frac{{\text{∂} {\varphi ^{(1)}}}}{{\text{∂} z}}\frac{{{\text{∂} ^2}{\varphi ^{(1)}}}}{{\text{∂} z\text{∂} t}}} \right) + \\& \qquad \qquad\qquad\qquad\frac{1}{g}\frac{{\text{∂} {\varphi ^{(1)}}}}{{\text{∂} t}}\left({g\frac{{{\text{∂} ^2}{\varphi ^{(1)}}}}{{\text{∂} {z^2}}} + \frac{{{\text{∂} ^3}{\varphi ^{(1)}}}}{{\text{∂} z\text{∂} {t^2}}}} \right).\end{split}$
Taking two distinct-frequency sinusoidal waves along the x-axis, the first order velocity potential due to the superposition is
$ \begin{split} & {\varphi ^{(1)}}(x,z,t) = \frac{{g{a_1}}}{{{\omega _1}}}\frac{{\cosh {k_1}(z + d)}}{{\cosh {k_1}d}} \sin ({k_1}x - {\omega _1}t) +\\ & \qquad\qquad\qquad \,\, \frac{{g{a_2}}}{{{\omega _2}}}\frac{{\cosh {k_2}(z + d)}}{{\cosh {k_2}d}}\sin ({k_2}x - {\omega _2}t) .\\ \end{split} $
Substituting Eq. (13) into Eq. (12) and on the surface, z=0, we have
$\begin{split}\frac{{{\text{∂} ^2}{\varphi ^{(2)}}}}{{\text{∂} {t^2}}} + & g\frac{{\text{∂} {\varphi ^{(2)}}}}{{\text{∂} z}} = \frac{1}{2}\sin 2{\theta _1}\left[ {3\frac{{{g^2}{a_1}^2{k_1}^2}}{{{\omega _1}}} ({{\tanh }^2}{k_1}d - 1)} \right] + \\ & \frac{1}{2}\sin 2{\theta _2} \left[ {3\frac{{{g^2}{a_2}^2{k_2}^2}}{{{\omega _2}}}({{\tanh }^2}{k_2}d - 1)} \right] + \\ & \frac{1}{2}\sin ({\theta _1} + {\theta _2})\left[ { - 2{g^2}{a_1}{a_2}{k_1}{k_2} \left( {\frac{1}{{{\omega _1}}} + \frac{1}{{{\omega _2}}}} \right) + } \right. \\ & 2{g^2}{a_1}{a_2}{k_1}{k_2} \tanh {k_1}d\tanh {k_2}d\left( {\frac{1}{{{\omega _1}}} + \frac{1}{{{\omega _2}}}} \right) - {g^2}{a_1}{a_2} \times \\& \left. {\left( {\frac{{{k_1}^2}}{{{\omega _1}}} + \frac{{{k_1}^2}}{{{\omega _2}}}} \right) + g{a_1}{a_2}({k_1}{\omega _1}\tanh {k_1}d + {k_2}{\omega _2}\tanh {k_2}d)} \right] + \\ & \frac{1}{2}\sin ({\theta _1} - {\theta _2}) \left[ { - 2{g^2}{a_1}{a_2}{k_1}{k_2}\left( {\frac{1}{{{\omega _1}}} - \frac{1}{{{\omega _2}}}} \right)} \right. + \\ & 2{g^2}{a_1}{a_2}{k_1}{k_2}\tanh {k_1}d\tanh {k_2}d\left( {\frac{1}{{{\omega _1}}} - \frac{1}{{{\omega _2}}}} \right) - \\ & {g^2}{a_1}{a_2}\left( {\frac{{{k_1}^2}}{{{\omega _1}}} - \frac{{{k_1}^2}}{{{\omega _2}}}} \right) + g{a_1}{a_2} \times \\& {({k_1}{\omega _1}\tanh {k_1}d - {k_2}{\omega _2}\tanh {k_2}d)} \bigg] ,\end{split} $
where ${\theta _1} = {k_1}x - {\omega _1}t$, and ${\theta _2} = {k_2}x - {\omega _2}t$.
Denote a function G where by:
$\begin{split} {G^ \pm }({\omega _1}, {\omega _2}) = & g_{12}^{^ \pm } \pm g_{_{21}}^{^ \pm }\\ = & - {g^2}\left[ {{k_1}{k_2}\left({\frac{1}{{{\omega _1}}} \pm \frac{1}{{{\omega _2}}}} \right)} \right. \times \\ & (1 \mp \tanh {k_1}d\tanh {k_2}d) + \\ & \left. {\frac{1}{2}\left({\frac{{{k_1}^2}}{{{\omega _1}{{\cosh }^2}{k_1}d}} \pm \frac{{{k_2}^2}}{{{\omega _2}{{\cosh }^2}{k_2}d}}} \right)} \right] .\end{split} $
Then
$\begin{split} \frac{{{\text{∂} ^2}{\varphi _2}}}{{\text{∂} {t^2}}} + g\frac{{\text{∂} {\varphi _2}}}{{\text{∂} z}} =& {a_1}{a_2}{G^ \pm }({\omega _1}, {\omega _2}) \sin ({k_ \pm }x - {\omega _ \pm }t) + \\ & \frac{1}{2}{a_1}^2{G^ + }({\omega _1}, {\omega _1})\sin 2({k_1}x - {\omega _1}t) + \\ & \frac{1}{2}{a_2}^2{G^ + }({\omega _2}, {\omega _2})\sin 2({k_2}x - {\omega _2}t),\end{split} $
where ${k_ \pm } = {k_1} \pm {k_2}$; and ${\omega _ \pm } = {\omega _1} \pm {\omega _2}$.
Consider a special solution:
$\begin{split} {\varphi ^{(2)}}(x, z, t) = & {a_1}{a_2}E_{12}^ \pm \frac{{\cosh {k_ \pm }(z + d)}}{{\cosh {k_ \pm }d}} \times \\ & \sin ({k_ \pm }x - {\omega _ \pm }t) + \frac{1}{2}{a_1}^2 {E_{11}^ \pm} \times \\ & \frac{{\cosh 2{k_1}(z + d)}}{{\cosh 2{k_1}d}}\sin 2({k_1}x - {\omega _1}t) + \\ & \frac{1}{2}{a_2}^2{E_{12}^ \pm} \frac{{\cosh 2{k_2}(z + d)}}{{\cosh 2{k_2}d}} \times \\ & \sin 2({k_2}x - {\omega _2}t) .\end{split}$
Besides the surface condition, this special solution should also satisfy the Laplace governing equations and the bottom boundary condition:
${\rm{for}}\; - d < z < 0, \;{\nabla ^2}{\phi ^{(2)}} = 0, $
${\rm{for}}\;z = - d, \frac{{\text{∂} {\phi ^{(2)}}}}{{\text{∂} z}} = 0.$
Substituting Eq. (17) into Eqs (18) and (19), we have
$\begin{split} E_{12}^ \pm = \frac{{{G^ \pm }({\omega _1}, {\omega _2})}}{{{D^ \pm }({\omega _1}, {\omega _2})}} \,\, E_{11}^ + = \frac{{{G^ + }({\omega _1}, {\omega _1})}}{{{D^ + }({\omega _1}, {\omega _1})}} \,\, E_{22}^ + = \frac{{{G^ + }({\omega _2}, {\omega _2})}}{{{D^ + }({\omega _2}, {\omega _2})}},\end{split} $
where ${D^ \pm }({\omega _1}, {\omega _2}) = g{k_ \pm }\tanh ({k_ \pm }d) - \omega _ \pm ^2$.
Utilizing the same principle, if an infinite number of wave components are considered, then the second order potential under the random wave condition can be obtained as follow:
$\begin{split} {\varphi ^{(2)}}(x,z,t) = & \sum\limits_i {\sum\limits_{j > i} {{a_i}{a_j}\frac{{{G^ \pm }({\omega _i},{\omega _j})}}{{{D^ \pm }({\omega _i},{\omega _j})}}} } \frac{{\cosh {k_ \pm }(z + d)}}{{\cosh {k_ \pm }d}} \times \\ &\sin ({k_ \pm }x - {\omega _ \pm }t + {\varepsilon _ \pm }) + \frac{1}{2}\sum\limits_i {{a_i}^2\frac{{{G^ + }({\omega _i},{\omega _i})}}{{{D^ + }({\omega _i},{\omega _i})}}} \times \\ & \frac{{\cosh 2{k_i}(z + d)}}{{\cosh 2{k_i}d}} \sin 2({k_i}x - {\omega _i}t + {\varepsilon _i}) .\end{split} $
For a random field of ocean waves, the surface displacement, velocity potential and Eulerian velocity to the second order are then given as
${\eta ^{(1)}}(x, t) = \sum\limits_i {{a_i}\cos ({k_i}x - {\omega _i}t + {\varepsilon _i})}, $
$\begin{gathered} {\phi ^{(1)}}(x, z, t) = \sum\limits_i {\frac{{{a_i}g}}{{{\omega _i}}}\frac{{\cosh {k_i}(z + d)}}{{\cosh {k_i}d}}} \times \\\qquad \quad \,\,\,\, \sin ({k_i}x - {\omega _i}t + {\varepsilon _i}) , \\ \end{gathered}$
$\begin{split} {\phi ^{(2)}}(x, z, t) = & \sum\limits_i {\sum\limits_{j > i} {{a_i}{a_j}\frac{{{G^ \pm }({\omega _i}, {\omega _j})}}{{{D^ \pm }({\omega _i}, {\omega _j})}}} } \times \\& \frac{{\cosh {k_ \pm }(z + d)}}{{\cosh {k_ \pm }d}} \sin ({k_ \pm }x - {\omega _ \pm }t + {\varepsilon _ \pm }) + \\ & \frac{1}{2}\sum\limits_i {{a_i}^2\frac{{{G^ + }({\omega _i}, {\omega _i})}}{{{D^ + }({\omega _i}, {\omega _i})}}\frac{{\cosh 2{k_i}(z + d)}}{{\cosh 2{k_i}d}}} \times \\& \sin 2({k_i}x - {\omega _i}t + {\varepsilon _i}) ,\end{split} $
$\begin{split} & {u^{(1)}}(x, z, t) = \frac{{\text{∂} {\phi ^{(1)}}}}{{\text{∂} x}} = \sum\limits_i {{a_i}{\omega _i}\frac{{\cosh {k_i}(z + d)}}{{\sinh {k_i}d}}} \times \\ & \qquad\qquad\qquad \cos ({k_i}x - {\omega _i}t + {\varepsilon _i}), \\ \end{split}$
$\begin{split} {u^{(2)}}(x, z, t) = & \sum\limits_i {\sum\limits_{j > i} {{a_i}{a_j}{k_ \pm }\frac{{{G^ \pm }({\omega _i}, {\omega _j})}}{{{D^ \pm }({\omega _i}, {\omega _j})}}} } \times \\ & \frac{{\cosh {k_ \pm }(z + d)}}{{\cosh {k_ \pm }d}}\cos ({k_ \pm }x - {\omega _ \pm }t + {\varepsilon _ \pm }) + \\& \sum\limits_i {{a_i}^2\frac{{{G^ + }({\omega _i}, {\omega _i})}}{{{D^ + }({\omega _i}, {\omega _i})}}\frac{{\cosh 2{k_i}(z + d)}}{{\cosh 2{k_i}d}}} \times \\ & {k_i}\cos 2({k_i}x - {\omega _i}t + {\varepsilon _i}) ,\end{split}$
$\begin{split} x - {x_0} = & \int\limits_0^t {u{\rm{d}}t \approx \sum\limits_i { - {a_i}\frac{{\cosh {k_i}({z_0} + d)}}{{\sinh {k_i}d}}} } \times \\ & \sin ({k_i}{x_0} - {\omega _i}t + {\varepsilon _i}) ,\end{split}$
$\begin{split} z - {z_0} = & \int\limits_0^t {\omega {\rm{d}}t} \cong \sum\limits_i {a\frac{{\cosh {k_i}({z_0} + d)}}{{\sinh {k_i}d}} \times } \\& \cos ({k_i}{x_0} - {\omega _i}t + {\varepsilon _i}),\end{split}$
where ${\omega _ \pm } = {\omega _i} \pm {\omega _j};{k_ \pm } = {k_i} \pm {k_j};{\varepsilon _ \pm } = {\varepsilon _i} \pm {\varepsilon _j}$; ${D^ \pm }({\omega _i}, {\omega _j}) = g{k_ \pm }$ $ \tanh ({k_ \pm }d) - {\omega _ \pm }^2$; and
$\begin{split}{G^ \pm }({\omega _i}, {\omega _j}) = & - {g^2}\left[ {\frac{{{k_i}{k_j}}}{{{\omega _i}{\omega _j}}}{\omega _ \pm }} \right. \times (1 \mp \tanh {k_i}d\tanh {k_j}d) + \\ & \left({\frac{{{k_i}^2}}{{2{\omega _i}{{\cosh }^2}{k_i}d}}} \pm {\left. { \frac{{{k_j}^2}}{{2{\omega _j}{{\cosh }^2}{k_j}d}}} \right)} \right]. \\ \end{split}$
Again, the Lagrangian horizontal velocity can be derived from the Taylor’s expansion:
$\begin{split}& u(x, z, t) = \sum\limits_i {{c_i}({a_i}{k_i})\frac{{\cosh {k_i}({z_0} + d)}}{{\sinh {k_i}d}}} \times \\& \qquad\qquad \cos ({k_i}{x_0} - {\omega _i}t + {\varepsilon _i}) + \\& \qquad\qquad \sum\limits_i {\sum\limits_{j > i} {{a_i}{a_j}{k_ \pm }\frac{{{G^ \pm }({\omega _i}, {\omega _j})}}{{{D^ \pm }({\omega _i}, {\omega _j})}}} } \times \\ & \qquad\qquad \frac{{\cosh {k_ \pm }({z_0} + d)}}{{\cosh {k_ \pm }d}} \times \\ & \qquad\qquad \cos ({k_ \pm }{x_0} - {\omega _ \pm }t + {\varepsilon _ \pm }) + \\& \qquad\qquad \sum\limits_i {{a_i}^2\frac{{{G^ + }({\omega _i}, {\omega _i})}}{{{D^ + }({\omega _i}, {\omega _i})}}} \frac{{\cosh 2{k_i}({z_0} + d)}}{{\cosh 2{k_i}d}}\; \times \end{split} $
$\begin{split}& \qquad\qquad \sum\limits_i {\sum\limits_j {\frac{{{a_i}{a_j}}}{2}} } \frac{{({k_i}{\omega _i} + {k_j}{\omega _j})}}{{\sinh {k_i}d \cdot \sinh {k_j}d}} \times \\& \qquad\qquad [\cos ({k_ - }{x_0} - {\omega _ - }t + {\varepsilon _ - })\cosh {k_ + }({z_0} + d) - \\& \qquad\qquad \cos ({k_ + }{x_0} - {\omega _ + }t + {\varepsilon _ + }) \cosh {k_ - }({z_0} + d)] + \\& \qquad\qquad \sum\limits_i {{c_i}} \frac{{{{({a_i}{k_i})}^2}}}{{2{{\sinh }^2}{k_i}d}}\cos 2({k_i}{x_0} - {\omega _i}t + {\varepsilon _i}) + \\& \qquad\qquad \sum\limits_i {{c_i}} {({k_i}{a_i})^2}\frac{{\cosh 2{k_i}({z_0} + d)}}{{2{{\sinh }^2}{k_i}d}} + O{(ka)^3} . \\ \end{split} $
In this expression, the first term denotes the first harmonic, the third and fifth terms the second harmonic, and the second and fourth terms the interaction among the different wave components. The last term is the drift velocity in the random wave field, which is simply the algebraic summation of the drift velocity from all the wave components, i.e.,
$\begin{gathered} {u_{\rm s}} = \sum\limits_{i = 1}^\infty {{c_i}{{({k_i}{a_i})}^2}\frac{{\cosh 2{k_i}({z_0} + d)}}{{2{{\sinh }^2}{k_i}d}}} \\ \; \; \; \quad \quad = 2\int\limits_0^\infty {s(\omega)\omega k\frac{{\cosh 2k({z_0} + d)}}{{2{{\sinh }^2}kd}}\operatorname{d} \omega }. \\ \end{gathered} $
In deep water condition, the surface drift at z0=0 would be
${u_{\mathop{\rm s}\nolimits} } = 2\int\limits_0^\infty {s(\omega )\omega k{\mathop{\rm d}\nolimits} \omega } ,$
which is identical to Eq. (1).
Equations (30) and (32) illustrate that when the wave energy spectrum is defined, the corresponding drift velocity would then be a time-independent quantity. In other words, the derivation implies an important outcome that the Stokes drift should not possess any fluctuation component, similar to the situation under the regular waves. Consequently, it should not cause any dispersion effect to the floating substances under the random wave field, which is contrary to the suggestion by Herterich and Hasselmann (1982) and others.
3 Experimental results and discussion
Less
In the above section, the derivations have shown that there should not be a dispersion effect to the floating substances due to the surface drift in random waves. To verify this conclusion, we carried out an experimental study as described in the following.
3.1 Experimental setup
Experiments were conducted in a wave flume measuring 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 the flume housed a two-dimensional piston type random wave-maker. The wave piston was controlled directly using the DHI active wave absorption control system (AWACS). Meanwhile, DHI wave synthesizer software was used to specify the types of wave required as well as their respective parameters. At the other end of the wave flume, an artificial absorbent beach was built to provide an efficient dissipation of the wave energy.
Four sensitive capacitance wave probes were mounted on a steel frame positioned at a range of 1.2–2.2 m from the wave paddle. The wave probes were capable of measuring wave heights to the nearest 0.5 mm. The probes had a small diameter (approximately 0.6 cm) such that their placement would not result in significant modifications of the wave profile.
Small vinyl marks with a radius of about 1 cm were used as floating particles in the experiments. The vinyl marks had a density of 0.9 g/cm3 and they were used to simulate the surface contaminant particles. They floated on the water surface without any relative movement to the surface water particles. A ruler is fixed on the wall of one side of the wave flume to determine the position of the markers. On the opposite side, a digital video camera mounted on a tripod was positioned beside the measurement area to capture the images of the particles’ movement. The field of view of the camera covered a span of about 2 m, in which the position of the small vinyl marks and their coordinates in the images can be identified. The schematic layout of the experimental set-up is shown in Fig. 2.
3.2 Experimental procedures
Two vinyl marks were placed on the water surface along the wave flume. According to Eq. (27), the first order horizontal displacement should be of the same order as the wave height (maximum Hs=0.05 m in this study). The initial distance for the two marks was thus set to be 0.2 m to avoid their collision. An experiment was initiated only after the water was sufficiently still. This can be judged by visually inspecting the motion of the marks for 1 min or 2 min. Any discernible movement would indicate the presence of a residual current. After it was ensured that there were no residual currents, the digital video camera was switched on and the wave generator was activated.
The experimental duration was chosen to be less than 100 s to avoid the effects by the reflected waves. Prior to the study, we had examined the effectiveness of the artificial beach installed in terms of dissipating the incoming waves. It was found that the beach was quite effective and the reflected wave height was less than 10% within the range of the wave conditions tested. Given the wave period of 1.0 s, the group velocity of the wave train would be equal to 0.78 m/s. The distance between the test span and the end of the flume was approximately 30 m. Hence, it would take about 80 s for the wave train to cover the distance to-and-fro. With the experimental duration less than 100 s, the reflected waves were thus not expected to have a significant effect.
3.3 Experimental results
In the experiments, the Pierson-Moscowitz (P-M) spectrum was used to generate the random waves. Three different tests were conducted with Hs =0.03, 0.04 and 0.05 m respectively with the water depth d=0.8 m. The measured wave spectra, plotted in Fig. 3, show that the spectrum generated in the tank agrees well with the theoretical P-M spectrum. In order to verify the repetition of experiments, each of the three tests were repeated three times, and similar results were obtained.
The distances ΔL between the marker particles determined by the coordinate-difference in the video recordings at different time is shown in Fig. 4. It can be observed that they only oscillate within a small range (±0.05 m) in these experiments during the test duration. These experimental observations are distinctly different from those of Buick et al. (2001) who reported that the relative distance of two marker particles increases with time. In their measurements, Buick et al. (2001) state that the turbulent diffusion have a significant effect, which implies that the wave in their experiments does not match potential flow anymore. However, the wave generated in the current study meets the linear wave criteria. This can be the reason leading to the difference.
The comparison between the experimental observations and Herterich and Hasselmann’s (1982) predictions is also presented in Fig. 4. Following Herterich and Hasselmann’s (1982) theory, after 100 s, the relative distance between the two markers would have increased by 8, 10 and 12 cm corresponding to the significant wave height 0.03, 0.04 and 0.05 m, respectively (see Appendix A). Clearly, this predicted increase was not observed in our experiments. The experimental results thus reinforce the analysis that the stochastic Stokes drift in the random waves does not lead to the surface dispersion of floating substances.
4 Conclusions
Less
The derivations of the Lagrangian velocities in both regular and random waves performed in this study illustrate that the Stokes drift velocity is a time-independent quantity that is among the components of the Lagrangian velocities, and thus it can also be considered as an instantaneous value. In this regards, the drift velocity should not possess any fluctuation once a wave spectrum is defined. Since the fluctuation is a necessary mechanism for the dispersion effect predicted by Herterich and Hasselmann (1982), the existence of the Stokes drift in the random waves should therefore not be able to induce a dispersion effect on surface floating substances. The experimental findings also verify this conclusion.
Funding
Less
  • The State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering Research Foundation of China under contract No. 2015491311.
References
Less
Buick J M, Morrison I G, Durrani T S, et al. 2001. Particle diffusion on a three-dimensional random sea. Experiments in Fluids, 30(1): 88–92
Craig P D, Banner M L. 1994. Modeling wave-enhanced turbulence in the ocean surface layer. Journal of Physical Oceanography, 24(12): 2546–2559
Dean R G, Dalrymple R A. 1991. Water Wave Mechanics for Engineers and Scientists. Singapore: World Scientific Publishing
Giarrusso C C, Carratelli Pugliese E, Spulsi G. 2001. On the effects of wave drift on the dispersion of floating pollutants. Ocean Engineering, 28(10): 1339–1348
Gōda Y. 2010. Random Seas and Design of Maritime Structures. 3rd ed. New Jersey: World Scientific Publishing Company, 416–479
Herterich K, Hasselmann K. 1982. The horizontal diffusion of tracers by surface waves. Journal of Physical Oceanography, 12(7): 704–711
Huang Chuanjian, Qiao Fangli, Wei Zexun. 2013. Effects of the surface wave-induced mixing on circulation in an isopycnal-coordinate oceanic circulation model. Acta Oceanologica Sinica, 32(5): 7–14
Longuet-Higgins M S. 1952. On the statistical distribution of the height of sea waves. Journal of Marine Research, 11(3): 245–266
Maksimenko N A. 1990. Comparative analysis of Lagrangian statistical characteristics for synoptic-scale currents in hydrophysical study areas. Oceanology, 30(1): 5–9
Mantovanelli A, Heron M L, Heron S F, et al. 2012. Relative dispersion of surface drifters in a barrier reef region. Journal of Geophysical Research, 117(C11): C11016
Mesquita O N, Kane S, Gollub J P. 1992. Transport by capillary waves: fluctuating stokes drift. Physical Review A, 45(6): 3700–3705
Pugliese Carratelli E, Dentale F, Reale F. 2011. On the effects of wave-induced drift and dispersion in the deepwater horizon oil spill. In: Liu Yonggang, Macfadyen A, Ji Zhengang, et al., eds. Monitoring and Modeling the Deepwater Horizon Oil Spill: A Record-Breaking Enterprise. Geophysical Monograph Series. Washington, DC: American Geophysical Union, 197–204
Sanderson B G, Pal B K. 1990. Patch diffusion computed from Lagrangian data, with application to the Atlantic equatorial undercurrent. Atmosphere-Ocean, 28(4): 444–465
US Army Corps of Engineers. 2002. Coastal Engineering Manual. USACE Publications, Philadelphia, US
Appendix
Less
Year 2018 volume 37 Issue 7
PDF
37
20
Cite this Article
BibTeX
Article Info
doi: 10.1007/s13131-018-1243-z
  • Receive Date：2017-07-08
  • Online Date：2026-04-14
  • Published：2018-07-25
Article Data
Affiliations
History
  • Received：2017-07-08
  • Accepted：2017-10-10
Funding
The State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering Research Foundation of China under contract No. 2015491311.
Affiliations
    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

Corresponding:

References
Share
https://castjournals.cast.org.cn/joweb/aos/EN/10.1007/s13131-018-1243-z
Share to
QR

Scan QR to access full text

Cite this article
BibTeX
Citations
表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
关闭全屏
  • BibTeX
  • EndNote
  • RefWorks
  • TxT
Links
Articles
Latest Articles
Most Read
Collections
Updates
Events
News
Multimedia
About
About Us
Contact
No. 86 Xueyuan South Road, Haidian District, Beijing
100081
010-62199257
qkjq@cast.org.cn

Copyright © 2025 China Association for Science and Technology. All rights reserved. For all open access content, the relevant licensing terms apply.
Sponsored by the Office of the Leading Group for Cybersecurity and Informatization of CAST, and supported by Science and Technology Review Publishing House