- Email: [email protected]
Mass transport in a two-layer wave boundary layer
Ocean Engineering 28 (2001) 1393–1411
Mass transport in a two-layer wave boundary layer Chiu-On Ng
*
Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong
Abstract The transport of a chemical species under the pure action of surface progressive waves in the benthic boundary layer which is loaded with dense suspended sediments is studied theoretically. The flow structure of the boundary layer is approximated by that of a two-layer Stokes boundary layer with a sharp interface between clear water and a heavy fluid. The simplest model of constant eddy diffusivities is adopted and the exchange of matter with the bed is ignored. For a thin layer of heavy fluid, whose thickness is comparable to the surface wave amplitude and the Stokes boundary layer thickness, effective transport equations are deduced using an averaging technique based on the method of homogenization. The effective advection velocity is found to be equal to the depth-averaged mass transport velocity, while the dispersion coefficient can be shown to be positive definite. Explicit expressions for the transport coefficients are obtained as functions of fluid properties and flow kinematics. Physical discussions on their relations are also presented. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Surface waves; Sediment suspension; Stokes boundary layer; Mass transport; Dispersion coefficient; Homogenization method
1. Introduction The sediment–water interface in the benthic boundary layer is a site where many interrelated physical, chemical and biological processes actively take place. Although the mechanisms are complex and many of them are still not well understood, these processes are believed to have a great impact on the quality of overlying water (McCave, 1974). Sea disposal of contaminated dredged spoil and solid wastes is now a common practice. The sedimentation of particles is often accompanied by * Tel.: +852-2859-2622; fax: +852-2858-5415. E-mail address: [email protected] (C.-O. Ng). 0029-8018/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 0 0 ) 0 0 0 5 7 - 3
1394
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
Nomenclature a b C Cs D d E G g H h Im i k LM N n Pe Re Sc s T t UE UI UL Um US u uI u∗ Vm v wf x y z z1 z2 b g
surface wave amplitude interface wave amplitude chemical concentration sediment concentration dispersion coefficient mud depth eddy diffusivity of mass complex parameter given by Eq. (51) acceleration due to gravity complex parameter given by Eq. (43) water depth the imaginary part of imaginary unit wavenumber the Monin–Obukhov length cell function defined by Eqs. (24) and (25) local vertical coordinate pointing from the mud bottom Peclet number the real part of Schmidt number relative density of sediment solid wave period time depth-averaged Eulerian streaming velocity amplitude of uI depth-averaged mass transport velocity amplitude of um1 depth-averaged Stokes drift horizontal component of fluid velocity near-bottom velocity given by the potential wave theory shear velocity amplitude of vm1 vertical component of fluid velocity falling velocity horizontal coordinate vertical coordinate complex number (=1⫺i) complex parameter (=zf+z∗) complex parameter (=zf⫺z∗) complex parameter given by Eq. (58) ratio of fluid density (=rw/rm)
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
⌬ d ⑀ z h n x r s f ⌿ y 具·典 (·) (·)1 (·)2 (·)m (·)w (·)∗ (·˜)
1395
Stokes boundary layer thickness for mass diffusion Stokes boundary layer thickness for momentum diffusion wave steepness ratio of Stokes boundary layer thicknesses (=dm/dw) surface displacement von Karman’s constant eddy viscosity interface displacement density wave angular frequency ratio of Stokes boundary layer thicknesses due to momentum and mass diffusion (=dm/⌬m) complex function given by Eq. (49) complex parameter given by Eq. (50) depth averaging time averaging first-order quantity second-order quantity quantity of mud quantity of water complex conjugate normalized quantity
continuous phase exchange of chemicals between sorbed and dissolved forms. It is possible that sorption of dissolved toxic materials such as polychlorinated biphenyls (PCBs), pesticides and heavy metals by suspended sediment, followed by deposition of the sediment, can scavenge substantial amounts of toxic material from the water column. Deposited materials, on the other hand, may interact with benthic organisms to result in diverse products. For example, bacterial degradation of organic matter may lead to disaggregation of sediments, nutrient regeneration, and so on, while microorganisms mediate many chemical reactions which may alter toxic chemicals to an even more toxic form. Input of man-made wastes may also cause diagenesis of natural sediment. On top of these, sediments are constantly in a dynamic state of motion owing to resuspension by waves, transport by currents, pumping by tides, stirring by bioturbation and so on, until burial deep below the interface by other deposits. When predicting the fate and transport of a chemical, one is often confronted with the task of working out the rates of the various processes that can occur on a specific interface. Though a comprehensive study is much desirable, the scope of a model is frequently limited by practical constraints. Simplifying a model is possible, however, only when one is able to estimate a priori which processes are dominant given a set of parameters in a particular geophysical setting. In this connection, a good
1396
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
quantitative tool, conceptual or mathematical, is an essential element. This is especially true for sites like the benthic boundary layer for which field data are scarce. The aim of this work was to examine specifically the mass transport in the benthic boundary layer which is laden with a heavy suspended load and is oscillatory under surface waves. While emphasizing only the physical forcings, the present work will be of great practical value in yielding relations by which one can estimate the transport rates of benthic sorbed contaminants under the sole influence of waves, which are an important agent in sediment transport on continental shelves. Transport of littoral sediment due to the combined action of waves and currents has been extensively studied (e.g., see Fredsøe and Deigaard, 1992; Nielsen, 1992; Madsen, 1993, and references therein). Typically, wave action is responsible for keeping sediments in suspension, while currents provide the net transport. Theoretical models of sediment transport where waves play a direct role in both aspects of entrainment and net transport have been presented by Mei and co-workers (Mei and Chian, 1994; Mei et al. 1997, 1998). They obtained effective advection–dispersion equations for transport of a finite cloud of low-concentration fine suspended sediments under the pure action of surface waves. The advection is solely due to waveinduced steady streaming, while dispersion is caused by shear in the wave boundary layer. Both processes are effective at a time scale much longer than a wave period. In the above-mentioned dispersion models, the sediment concentration is assumed to be so low that the flow characteristics are essentially not affected by the particles. This assumption, however, breaks down when the solid concentration becomes a finite fraction of the fluid density. One direct effect of the presence of solid particles is to dampen the turbulent fluctuations, as can be inferred from the variation of von Karman’s constant with sediment concentration. The constant drops from its value of 0.4 in clear fluid to as low as 0.15 for a sufficiently heavy suspended load (Einstein and Chien, 1954). Einstein and Chien (1955) further advanced a modified velocity distribution equation for steady sediment-laden stream flows. On fitting with experimental data, they found it necessary to divide the flow into two zones. Close to the bed is a shallow layer (thickness being of the order of one-tenth of the total depth) of a heavy-fluid zone where the sediment is heavily concentrated. Overlying this zone is a light-fluid zone where the sediment concentration is small and does not change the fluid density. The coarser the particles, the larger the fraction of the solid population residing in the heavy-fluid zone. Therefore, in a concentrated sedimentladen flow, the heavy-fluid zone, because of its much larger solid–water ratio, will be responsible for transportation of most of the sorbed chemicals. The intention of the present work was to develop analytically a transport model for contaminated particles, which may originate from human dumping, in such a heavily loaded boundary layer under the pure action of progressive waves. The theory of Mei and Chian (1994) is extended to a two-layer wave boundary layer, based on the two-fluid-zone structure of Einstein and Chien (1955). We study a simplified problem in order to capture the essence of the physics. One of the idealizations is to model the abrupt yet continuous stratification by two distinct layers of homogeneous fluid: the upper fluid is clear water while the lower fluid is a denser liquid
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1397
suspension. By this construction, a sharp interface between the two fluids is assumed to persist. This will not be too strong an assumption in practice as long as the stratification is stable, as should be in the present case. If the bottom layer thickness is comparable to the Monin–Obukhov length, the layer will be well-mixed because the turbulence will prevail over buoyancy forces within a layer of this scale. Consequently vertical diffusion of mass may occur within the layer, but diffusion across the density interface that forms at the top of the stable layer is strongly inhibited. Momentum transport, on the other hand, can be transmitted across the density interface by internal waves. It has been reported that some high-density fluid mud suspensions appear to be stable, even under large shearing stresses (McCave, 1974). Our problem is described further in Section 2. Mass transport of a chemical takes place in the denser fluid of a two-layer wave boundary layer (also known as the Stokes boundary layer) under small-amplitude surface progressive waves. The basic governing equations and the assumptions regarding the scalings are first stated in Section 2 before variables are expanded in terms of the small wave steepness ⑀. This is followed by a perturbation analysis in Section 3. The analysis is based on the method of homogenization, which has been utilized to obtain effective macroscale equations for a wide range of flow and transport problems (Mei et al., 1996). In this case, effective transport equations are deduced up to O(⑀2), in which the advection velocity and dispersion coefficient show up as functions of fluid properties and flow characteristics. It will be shown that the advection velocity is equal to the depthaveraged mass transport velocity or Lagrangian drift, while the dispersion coefficient is always positive. Explicit expressions for the advection velocity and the dispersion coefficient are then deduced in Sections 4 and 5 respectively upon invoking the results of Ng (2000) and solving a canonical “cell” problem. Physical discussions are then presented in Section 6 to illustrate the dependence on various parameters of these transport coefficients, in dimensional and normalized forms.
2. Formulation As shown in Fig. 1, a train of two-dimensional, small-amplitude progressive waves is propagating on the surface of a clear water body of uniform depth h, which is overlying a thin, heavy-fluid layer of thickness d above the bottom. The bottom is assumed to be flat, impermeable and rigid. In other words, effects due to bedform and exchange of particles with the bed are not taken into consideration. The heavy fluid, which is essentially a well-mixed suspension, will also be referred to as “mud” for simplicity. The flows are turbulent. Like Mei and Chian (1994), we assume that near the bottom the eddy viscosities and diffusivities are constants in order to facilitate analytical development. Such an approximation is commonly employed in oceanographical studies and is believed to be good enough in many practical cases. Also, a sharp interface between the fluids is assumed, as discussed earlier. A twolayer Stokes boundary layer model that has recently been developed by Ng (2000) will be followed in this work. In an (x, y) coordinate system where x is the still water level and y is pointing
1398
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
Fig. 1.
Sketch of problem geometry.
vertically upward, the surface profile for progressive waves propagating in the positive x-direction is given by y⫽h(x, t)⫽Re{a ei(kx−st)},
(1)
where a is the wave amplitude, i is the imaginary unit, t is time, k is the wavenumber and s is the wave angular frequency. Without loss of generality, a and s are assumed to be real. The flow is essentially inviscid except very near the boundaries. At the leading order, k is also real and governed by the dispersion relation s2⫽gk tanh kh,
(2)
where g is the acceleration due to gravity. The near-bottom velocity given by the inviscid flow theory is uI⫽Re{UI ei(kx−st)}
sa where UI⫽ . sinh kh
(3)
The vertical extents of the bottom turbulent diffusion of momentum and mass are scaled by the corresponding Stokes boundary layer thicknesses: dm⬅(2nm/s)1/2 and ⌬m⬅(2Em/s)1/2,
(4)
where nm and Em are respectively the eddy diffusivity of momentum and mass. Typically the Schmidt number Sc⬅nm/Em is of order unity (supported by Reynolds analogy), and therefore the two Stokes boundary layer thicknesses are in general comparable to each other. We further assume that they are also of the same order of magnitude as the undisturbed mud depth d and the wave amplitude a, which are all much smaller than the wavelength 2p/k and the water depth h. The small wave steepness ⑀⬅ka苲kd苲kdm苲k⌬m1
(5)
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1399
will be used as an ordering parameter. For good mixing within the mud layer while maintaining a stable density interface on the top, the thickness of the layer d must also be in the order of the Monin– Obukhov length for suspended sediment stratification, which is given by LM⫽u3∗/(s⫺1)gwfCs,
(6)
where u∗ is the shear velocity, s is the specific weight of sediment solid, is von Karman’s constant, wf is the falling velocity of sediments and Cs is the volumetric concentration of sediments. To describe flow and transport near the bottom, it is more convenient to use a local vertical coordinate n⬅y+h+d, which points upward from the bottom of the mud layer (Fig. 1). The mud–water interface is given by n=d+x where x(x, t)⫽Re{b ei(kx−st)}
(7)
is the interface displacement with a complex amplitude b, whose modulus is an order of magnitude smaller than a. Let C be the concentration of a chemical in the mud (i.e., total mass of a chemical species, say in sorbed form, per unit volume of the heavy fluid) and (um, vm) be the horizontal and vertical components of the mud fluid velocity. Then the mass transport of the chemical is governed by
冉
冊
∂C ∂C ∂2C ∂2C ∂C ⫹⑀um ⫹⑀vm ⫽Em ⑀2 2 ⫹ 2 , ∂t ∂x ∂n ∂x ∂n
(8)
where the eddy diffusivity of mass is assumed to be the same in horizontal and vertical directions. Since mass exchanges with the bed and the overlying clear water are ignored, the normal flux at these interfaces must be zero: Em
∂C ⫽0 on n⫽0 ∂n
Em
冉
and
(9)
冊
∂C x 2 ∂3 C ∂2C ⫹O(⑀3)⫽0 on n⫽d. ⫹⑀x 2 ⫹⑀2 ∂n ∂n 2 ∂n3
(10)
The kinematic condition at the mud–water interface is ∂x ⫽v ⫹O(⑀) on n⫽d. ∂t m
(11)
Note that, in Eqs. (10) and (11), the conditions at the exact position n=d+x have been approximated by those at n=d using Taylor series expansions. Also, the small ordering parameter ⑀ has been inserted in Eqs. (8)–(10) to indicate the relative order of magnitude of the associated term, as previously shown by Mei and Chian (1994). These orders are based on the following scalings: x=O(k⫺1), n=O(d)=O(⌬m)=O(⑀k⫺1), t=O(s⫺1), um=O(UI), vm=O(⑀UI), x=O(⑀a) and the Peclet number Pe⬅UI/kEm=O(⑀⫺1). In the absence of steady currents at the leading order, the pri-
1400
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
mary time scale t2 over which wave-induced transport becomes effective is that due to spreading by dispersion over one horizontal length scale, which is two orders of magnitude longer than a wave period since t2/t=O(s/Emk2)=O(⑀⫺2). Let us now expand the variables and the time derivative: C→C0⫹⑀C1⫹⑀2C2⫹O(⑀3),
(12)
(um, vm)→(um1, vm1)⫹⑀(um2, vm2)⫹O(⑀2),
(13)
∂/∂t→∂/∂t⫹⑀ ∂/∂t2.
(14)
2
The leading order velocity components can be further expanded (Ng, 2000): (um1, vm1)⫽Re{(Um, Vm) ei(kx−st)},
(15)
where Um(n) and Vm(n) are related to each other by the continuity equation: ikUm⫽⫺V⬘m.
(16)
A relation for the amplitude of the interface displacement is also found by substituting Eqs. (7) and (15) into Eq. (11): b⫽is−1Vm(d).
(17)
Note that the second-order velocity um2 has a steady component, known as the Eulerian streaming velocity um2, which will also appear in later deductions.
3. Effective transport equations Successive orders of perturbation equations are obtainable by substituting Eqs. (12)–(14) into Eqs. (8)–(10), which are discussed as follows. The problem at O(1) is ∂2C0 ∂C0 ⫽Em 2 ∂t ∂n
in 0⬍n⬍d
(18)
and ∂C0 ⫽0 on n⫽0, d. ∂n
(19)
The leading order concentration C0 is taken to represent the smooth component whose variations with the short time t die out in the long term (Mei and Chian, 1994; Mei et al., 1996). Hence it is clear that, on ignoring the transience, C0 is independent of n as well, or C0⫽C0(x, t2).
(20)
On simplifying using Eq. (20) and substituting Eq. (15), the O(⑀) equation becomes
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1401
∂C0 ∂C1 ∂2C1 ⫹Re{Um ei(kx−st)} ⫽Em 2 in 0⬍n⬍d, ∂t ∂x ∂n
(21)
which is subject to the boundary conditions ∂C1 ⫽0 on n⫽0, d. ∂n
(22)
By linearity of Eq. (21), the following form for C1 is suggested: ∂C0 , C1(x, n, t)⫽Re{N(n) ei(kx−st)} ∂x
(23)
where the function N(n) is governed by Em
d2N ⫹isN⫽Um dn2
in 0⬍n⬍d
(24)
and dN ⫽0 on n⫽0, d. dn
(25)
Solutions of N(n) will be sought in Section 5 when we derive expressions for the dispersion coefficient. Let us now consider equations of O(⑀2). In a conservative form, the perturbation equation of this order reads
冉
∂ ∂C0 ∂2C0 ∂2C2 ∂C0 ∂C2 ∂ ⫹ ⫹ (um1C1)⫹ (vm1C1)⫹um2 ⫽Em ⫹ ∂t2 ∂t ∂x ∂n ∂x ∂x2 ∂n2
冊
in 0⬍n
(26)
⬍d, while the boundary conditions are ∂C2 ⫽0 on n⫽0 ∂n
(27)
∂C2 ∂2C1 ⫹x 2 ⫽0 on n⫽d. ∂n ∂n
(28)
and
Taking the depth average followed by the time average of Eq. (26) while using the boundary conditions yields
冋
1 ∂2C1 ∂C0 ∂ ⫹ 具um1C1典⫹ vm1C1⫹Emx 2 ∂t2 ∂x d ∂n
册
⫹具um2典 n=d
∂C0 ∂2C0 ⫽Em 2 , ∂x ∂x
(29)
where the angle brackets and the overbar denote the depth average and the time average over a wave period T, respectively:
1402
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
冕 d
冕
t⫹T
1 1 具f典⬅ f dn, ¯f⬅ d T 0
f dt.
(30)
t
On substituting Eqs. (15, 7) and (23), the second and third terms on the left-hand side of Eq. (29) can be developed as follows: 1 ∂2C0 ∂ 具um1C1典⫽ Re具UmN∗典 2 , ∂x 2 ∂x
(31)
where the asterisk denotes the complex conjugate, and
冋
册 冋 冉
∂2C1 1 vm1C1⫹Emx 2 d ∂n ⫽
冋
册
1 d2N∗ ⫽ Re VmN∗⫹Emb 2 dn n=d 2d
冊册
d2N∗ 1 Re is−1Vm Em 2 ⫺isN∗ 2d dn
∂C0 n=d ∂x
(32)
∂C0 ∂C0 1 ⫽ Re[is−1Vm(d)U∗m(d)] , ∂x 2d ∂x n=d
where Eqs. (17) and (24) have been used in Eq. (32). Putting these back to Eq. (29), we finally have the effective transport equation: ∂2C0 ∂C0 ∂C0 ⫹UL ⫽[Em⫹D] 2 , ∂t2 ∂x ∂x
(33)
where UL⫽UE⫹US,
(34)
in which UE⫽具um2典
(35)
is the depth-averaged Eulerian streaming velocity, US⫽
1 Re[is−1Vm(d)U∗m(d)] 2d
(36)
is an additional component of the advection velocity, and 1 D⫽⫺ Re具UmN∗典 2
(37)
is the dispersion coefficient. Eq. (33) governs the transport of a chemical species in the heavy-fluid layer, in which the wave action alone induces both advection and dispersion, which become effective over a long time scale T2=O(⑀⫺2T). Some further analytical relationships can be developed as follows. We first show that the additional advection velocity component US is actually equal to the depth-averaged Stokes drift. For the present two-dimensional problem, Stokes drift (Longuet-Higgins, 1953) is given by:
冉冕 冊 冉冕 冊 um1 dt
∂um1 ⫹ ∂x
vm1 dt
∂um1 ⫽Re{is−1Um ei(kx−st)}Re{⫺V⬘m ei(kx−st)} ∂n
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1 ∗ ⬘ ⫹Re{is−1Vm ei(kx−st)}Re{U⬘m ei(kx−st)}⫽ Re{is−1(Um Vm⫹VmU∗⬘ m )} 2
1403
(38)
1 ⫽ Re{is−1(VmU∗m)⬘}, 2 whose depth average is clearly equal to US: 1 1 ∗ ∗ Re具is−1(VmUm )⬘典⫽ Re[is−1Vm(d)Um (d)]⫽US. 2 2d
(39)
Hence in Eq. (33) the advection velocity, composed of Eulerian streaming velocity and Stokes drift, amounts to the depth-averaged Lagrangian drift or mass transport velocity of individual fluid particles. We next show that D is always positive, which is a necessary condition for a physically admissible diffusion coefficient. On substituting Eq. (24) for Um into Eq. (37), we may obtain in general
再
Em Em 1 D⫽⫺ Re具(EmN⬙⫹isN)N∗典⫽⫺ Re具N∗N⬙典⫽⫺ Re [N∗N⬘]d0 2 2 2d
冕 d
(40)
冎 冕 d
Em 兩N⬘兩2 dn⬎0, ⫺ N⬘N⬘∗ dn ⫽ 2d 0
0
where integration by parts and the boundary conditions (25) have been used.
4. Advection velocity Let us now derive explicit expressions for the mass transport velocity. From Ng (2000), the following relations may be invoked: Um(n˜ )⫽(g⫺g cosh zn˜ ⫹H sinh zn˜ )UI
(41)
z Vm(n˜ )⫽ kdm[g(zn˜ ⫺sinh zn˜ )⫹H(cosh zn˜ ⫺1)]UI, 2
(42)
and
where H⫽
g(1−g)+g2 cosh zd˜ +gz sinh zd˜ . z cosh zd˜ +g sinh zd˜
(43)
In Eqs. (41) and (42), the following definitions of notation have been used: z⬅1⫺i is a complex constant,
(44)
1404
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
(n˜ , d˜ )⬅(n, d)/dm
(45)
are the normalized vertical coordinate and mud depth with respect to the momentum Stokes boundary layer thickness, g⬅rw/rm
(46)
is the ratio of the density of water (rw) to that of mud (rm), and z⬅dm/dw⫽(nm/nw)1/2
(47)
is the ratio of the momentum Stokes boundary layer thickness in mud (dm) to that in water (dw), where nw is the eddy viscosity of water. The Eulerian streaming velocity has also been found by Ng (2000): um2(n˜ )⫽ks−1U2IRe{⌿(n˜ )⫹yn˜ ⫹3g2/4},
(48)
where ⌿(n˜ )⫽(1/8)(g2⫹兩H兩2) cosh 2n˜ ⫹(1/8)(g2⫺兩H兩2) cos 2n˜ ⫺(g/8)(H ⫹H∗) sinh 2n˜ ⫹(ig/8)(H⫺H∗) sin 2n˜ ⫺(z∗gH∗/2)n˜ cosh z∗n˜ ⫺(g2
(49)
⫺i兩H兩 /2) cosh z n˜ ⫹(z g /2)n˜ sinh z n˜ ⫹(gH ⫺igH/2) sinh z n˜ 2
∗
∗ 2
∗
∗
∗
and y⫽⫺(1/4)(g2⫹兩H兩2) sinh 2d˜ ⫹(1/4)(g2⫺兩H兩2) sin 2d˜ ⫹(g/4)(H ⫹H∗) cosh 2d˜ ⫺(ig/4)(H⫺H∗) cos 2d˜ ⫹(z∗g2/2⫹igH∗d˜ ⫹z兩H兩2/2) sinh z∗d˜ ⫺(z∗gH∗/2⫹ig2d˜ ⫹zgH/2) cosh z∗d˜ ⫺(z∗gG∗/2z)[z∗zgd˜ ⫺igz sinh zd˜
(50)
⫹izH(cosh zd˜ ⫺1)⫹1⫹iG]⫺g兩G兩 /2z⫺(z/2)[g(zd˜ ⫺sinh zd˜ )⫹H(cosh zd˜ ⫺1)][g cosh z∗d˜ ⫺H∗ sinh z∗d˜ ⫹gG∗], 2
in which G⫽
−gz−(1−g)z cosh zd˜ . z cosh zd˜ +g sinh zd˜
(51)
Also, the depth average of ⌿ is 具⌿典⫽(1/16d˜ )(g2⫹兩H兩2) sinh 2d˜ ⫹(1/16d˜ )(g2⫺兩H兩2) sin 2d˜ ⫺(g/16d˜ )(H ⫹H∗)(cosh 2d˜ ⫺1)⫺(ig/16d˜ )(H⫺H∗)(cos 2d˜ ⫺1)⫹(g2/2⫹3zgH∗/4d˜ ⫺z∗gH/4d˜ ) cosh z∗d˜ ⫺(gH∗/2⫹3zg2/4d˜ ⫺z∗兩H兩2/4d˜ ) sinh z∗d˜ ⫺3zgH∗/4d˜
(52)
⫹z∗gH/4d˜ . Hence, the depth-averaged Eulerian streaming velocity is given by UE⫽具um2典⫽ks−1U2IRe{具⌿典⫹yd˜ /2⫹3g2/4},
(53)
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1405
while the depth-averaged Stokes drift is, from Eq. (36), kU2I Re{(zd˜ )−1[g(zd˜ ⫺sinh zd˜ )⫹H(cosh zd˜ ⫺1)][g⫺g cosh z∗d˜ 2s ⫹H∗ sinh z∗d˜ ]}.
US⫽
(54)
5. Dispersion coefficient To obtain explicit expressions for the dispersion coefficient, we first need to solve for N from Eq. (24), which after substituting Eq. (41) can be written as UId2m d 2N ⫺(zf)2N⫽(g⫺g cosh zn˜ ⫹H sinh zn˜ ) , 2 dn˜ Em
(55)
where f⫽dm/⌬m⫽(nm/Em)1/2⫽Sc1/2
(56)
is the ratio of the Stokes boundary layer thicknesses due to momentum and mass diffusion, and is equal to the square root of the Schmidt number. As discussed earlier, this ratio should be of order unity. We however assume that f⫽1 for generality and may obtain the following solution satisfying Eqs. (55) and (25) id2mUI [⫺H sinh zfn˜ ⫹b cosh zfn˜ ⫹fH sinh zn˜ ⫺fg cosh zn˜ N(n˜ )⫽ 2f(1−f2)Em
(57)
⫺g(1⫺f2)/f], where b⫽
H(cosh zfd˜ −cosh zd˜ )+g sinh zd˜ . sinh zfd˜
(58)
We may now plug Eqs. (41) and (57) into Eq. (37), and obtain after some algebra: D⫽
再
(gH+bH∗)[cosh z1d˜ −1]−(gb+兩H兩2) sinh z1d˜ fs−1U2I Im 4(1−f2)d˜ z1
(59)
冎
(gH−bH∗)[cosh z2d˜ −1]−(gb−兩H兩2) sinh z2d˜ , ⫹ z2 where z1⫽zf⫹z∗,
z2⫽zf⫺z∗.
(60)
Using L’Hospital’s rule, one may show that the dispersion coefficient tends to the following finite limit when f approaches unity:
1406
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
s−1U2I {Re([g2 sinh z∗d˜ ⫹gH∗(cosh zd˜ ⫺cosh z∗d˜ )]/sinh zd˜ ) lim D⫽ 8 f→1 ⫹(1/4d˜ )(g2⫹兩H兩2) sinh 2d˜ ⫺(1/4d˜ )(g2⫺兩H兩2) sin 2d˜ ⫺(gHr/2d˜ )(cosh 2d˜ ⫺1) ⫺(gHi/2d˜ )(cos 2d˜ ⫺1)⫺兩H兩2},
(61)
where Hr and Hi are respectively the real and imaginary parts of H.
6. Physical discussions Explicit expressions have now been obtained for the effective advection velocity UL=UE+US, and the dispersion coefficient D. These are functions of the wave frequency s, near-bottom velocity UI (which depends on the wave amplitude, wavenumber k and water depth h), and the following ratios: g=rw/rm, z=(nm/nw)1/2, f=(nm/Em)1/2 and d˜ =d/dm. Clearly, a heavy-fluid suspension is denser than water, so the density ratio must be less than unity: g⬍1. Also, zⱕO(1) because an eddy viscosity for wall turbulence typically increases with the distance from the wall, and also the mixing length for momentum in clear water should be greater than that in a particle-laden liquid. In general, the parameter z controls the relative rates of momentum diffusion across the mud and the water boundary layer. The ratio f, or the square root of the Schmidt number, is equal to unity by Reynolds analogy. However, this number can be different from unity since the mixing length for mass, because of inertia, is somewhat shorter than that for momentum; but, on the other hand, solid particles may be thrown further to the outside of eddies by centrifugal forces (Vanoni, 1975). Some sample values of the transport coefficients have been computed and are shown in Table 1 for the cases h=20 and 40 m and T=5, 10, 15 and 20 s, where a=0.5 m, g=0.75, z=0.5, f=1.0 and d˜ =2.0. The wave periods are that typical of wind waves. It is clear that the velocities and the dispersion coefficient increase with the near-bottom velocity UI, which in turn is greater for longer waves over a shallower water layer. Except for rather short waves, the mass transport velocity is in the order of 1 cm/s (Eulerian streaming velocity and Stokes drift being comparable to each other), while the dispersion coefficient is the range of O(1)–O(100) cm2/s. These values are at least of the same order of magnitude as that due to tidal currents, and confirm the importance of the action of wind waves on the transport of matter in the benthic boundary layer. Of course, for a deep water layer or large kh, these waveinduced transport mechanisms can become subdominant. The variations of the transport coefficients, in dimensionless form, with the independent parameters are further examined in Figs. 2–5. Non-dimensional coefficients are defined as follows: ˜ S, U ˜ L)⬅(UE, US, UL)/ks−1UI, D ˜ ⫽D/s−1U2I. ˜ E, U (U
(62)
˜ L (solid lines) and U ˜ E (dashed lines) as functions of d˜ for various g Fig. 2 shows U and z. The value of Stokes drift can be obtained from the difference of the corre-
0.161 3.23 0.050 0.18 0.18 0.36 0.35
5
T (s)
k (m⫺1) kh UI (m/s) UE (cm/s) US (cm/s) UL (cm/s) D (cm2/s)
20
h (m)
0.052 1.04 0.255 0.60 0.57 1.17 18.1
10 0.032 0.64 0.308 0.67 0.64 1.31 39.7
15 0.023 0.46 0.327 0.69 0.66 1.35 59.5
20
5 0.161 6.44 0.002 0.007 0.007 0.014 0.0006
40
0.043 1.72 0.117 0.23 0.22 0.45 3.79
10
0.024 0.96 0.188 0.31 0.30 0.60 14.7
15
0.017 0.68 0.214 0.33 0.32 0.65 25.6
20
Table 1 Sample model values of velocities and dispersion coefficient under typical wind-wave conditions, for a=0.5 m, g=0.75, z=0.5, f=1.0 and d˜ =2.0
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411 1407
1408
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
˜ L (solid lines) and Eulerian streaming veloFig. 2. Normalized depth-averaged mass transport velocity U ˜ E (dashed lines) as functions of d˜ and g, for (a) z=0.1 and (b) z=1.0. city U
Fig. 3.
˜ as function of f and d˜ , for g=z=0.5. Normalized dispersion coefficient D
sponding pair of curves. Clearly the advection velocity components increase with d˜ and g, or the advection rate is higher for a thicker layer of lighter mud. This is reasonable because for such a mud layer more fluid elements can be induced into more appreciable motion by the waves. The motion of denser mud tends to be more sluggish under wave action (Ng, 2000). An increase in z will however lower the velocities, especially for smaller d˜ and g. This is so because a larger z for a fixed nm means a smaller nw or a smaller rate of momentum diffusion across the water boundary layer. ˜ is shown in Fig. 3, where g=z=0.5 are assumed. The The effect of f=Sc1/2 on D dispersion coefficient reaches a maximum at a value of f which is of the order of unity, the specific values being dependent on the mud layer thickness d˜ . For instance,
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1409
Fig. 4.
˜ as function of d˜ and g, for z=0.1 and (a) f=1.0 and (b) f=2.0. Normalized dispersion coefficient D
Fig. 5.
˜ as function of d˜ and g, for z=1.0 and (a) f=1.0 and (b) f=2.0. Normalized dispersion coefficient D
˜ is the maximum at f=1.1 when d˜ =2, but at f=2.2 when d˜ =1. The assumption of D Reynolds analogy or Sc=1 will in general lead to a dispersion coefficient that is closer to the peak value for a thicker mud layer. ˜ increases with d˜ until a maximum is reached, Figs. 4 and 5 further show that D ˜ and the corresponding d˜ being functions of the parathe values of the maximum D ˜ occurs when meters g, z and f. For the cases shown in the plots, the maximum D d˜ is in the range 1–2.5. Also, a lower g or a denser mud layer will correspond to a smaller dispersion coefficient, the effect being more prominent for a larger z. Increas˜ to occur at a smaller d˜ . In general, one ing the value of f will cause the peak D may expect that the dispersion coefficient is near its peak value when the mud depth is one to two times the Stokes boundary layer thickness. The dispersion coefficient will drop significantly as the mud layer becomes too thin or too thick as compared with the Stokes boundary layer thickness.
1410
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
7. Concluding remarks In this work, we have presented an analytical model for transport of a chemical species sorbed on to suspended sediments in a heavily loaded benthic boundary layer under the pure action of surface progressive waves. The key assumptions are: (1) the stratified boundary layer is modelled by a two-layer boundary layer with a sharp interface; (2) the heavy-fluid layer has a thickness comparable to the Monin–Obukhov length, the Stokes boundary layer thicknesses and the wave amplitude; (3) fluid properties including eddy viscosity and diffusivity are constant within a fluid layer; and (4) bedform and exchange with the bed are ignored. Following an asymptotic method of averaging which is based on the homogenization technique, we have deduced an effective transport equation, as formally given by Eqs. (33)–(37). The effective advection velocity is found to be equal to the depth average of the mass transport velocity, comprising Eulerian streaming velocity and Stokes drift. Also, the dispersion coefficient can be formally shown to be positive definite. Explicit expressions for these transport coefficients are given in Eqs. (53, 54) and (59) as functions of characteristics of the waves and the boundary layer. We have also shown that the benthic advection and dispersion rates due to typical wind waves alone can be as significant as those due to tidal currents. The dispersion coefficient is found to reach a maximum when the depth of the heavy fluid is slightly thicker than the Stokes boundary layer thickness, and when the Schmidt number is slightly larger than unity. Despite the simplifying assumptions, the present theory provides one with an analytical tool to estimate the effects of pure waves on the transport of a benthic species. Of course, other factors that have not been taken into consideration may be just as influential as the waves. A more complete model may have the individual components due to various factors lumped into the advection and dispersion coefficients. Our theory can offer a modular component due to pure waves. Future efforts to advance a more comprehensive transport model are deemed worthwhile.
Acknowledgements The work described in this paper was jointly supported by two grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Projects HKU 7117/99E and NSFC/HKU 8).
References Einstein, H.A., Chien, N., 1954. Second approximation to the solution of the suspended-load theory. Report No. 3. University of California, Institute of Engineering Research. Einstein, H.A., Chien, N., 1955. Effects of heavy sediment concentration near the bed on velocity and sediment distribution. Report No. 8. University of California, Institute of Engineering Research. Fredsøe, J., Deigaard, R., 1992. Mechanics of Coastal Sediment Transport. World Scientific, Singapore. Longuet-Higgins, M.S., 1953. Mass transport in water waves. Phil. Trans. R. Soc. Lond. A245, 535–581.
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1411
Madsen, O.S., 1993. Sediment Transport on the Shelf. Parsons Lab., MIT, Cambridge, MA. McCave, I.N. (Ed.), 1974. The Benthic Boundary Layer. Plenum Press, New York. Mei, C.C., Chian, C., 1994. Dispersion of small suspended particles in a wave boundary layer. J. Phys. Oceanogr. 24, 2479–2495. Mei, C.C., Auriault, J.L., Ng, C.O., 1996. Some applications of the homogenization theory. In:. Hutchinson, J.W., Wu, T.Y. (Eds.), Advances in Applied Mechanics, vol. 32. Academic Press, California, pp. 277–348. Mei, C.C., Fan, S.J., Jin, K.R., 1997. Resuspension and transport of fine sediments by waves. J. Geophys. Res. 102 (C7), 15807–15821. Mei, C.C., Chian, C., Ye, F., 1998. Transport and resuspension of fine particles in a tidal boundary layer near a small peninsular. J. Phys. Oceanogr. 28 (11), 2313–2331. Ng, C.O., 2000. Water waves over a muddy bed: a two-layer Stokes’ boundary layer model. Coastal Eng. 40 (3), 221–242. Nielsen, P., 1992. Coastal Bottom Boundary Layers and Sediment Transport. World Scientific, Singapore. Vanoni, V.A., 1975. Sedimentation Engineering. American Society of Civil Engineering, New York.
Mass transport in a two-layer wave boundary layer Chiu-On Ng
*
Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong
Abstract The transport of a chemical species under the pure action of surface progressive waves in the benthic boundary layer which is loaded with dense suspended sediments is studied theoretically. The flow structure of the boundary layer is approximated by that of a two-layer Stokes boundary layer with a sharp interface between clear water and a heavy fluid. The simplest model of constant eddy diffusivities is adopted and the exchange of matter with the bed is ignored. For a thin layer of heavy fluid, whose thickness is comparable to the surface wave amplitude and the Stokes boundary layer thickness, effective transport equations are deduced using an averaging technique based on the method of homogenization. The effective advection velocity is found to be equal to the depth-averaged mass transport velocity, while the dispersion coefficient can be shown to be positive definite. Explicit expressions for the transport coefficients are obtained as functions of fluid properties and flow kinematics. Physical discussions on their relations are also presented. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Surface waves; Sediment suspension; Stokes boundary layer; Mass transport; Dispersion coefficient; Homogenization method
1. Introduction The sediment–water interface in the benthic boundary layer is a site where many interrelated physical, chemical and biological processes actively take place. Although the mechanisms are complex and many of them are still not well understood, these processes are believed to have a great impact on the quality of overlying water (McCave, 1974). Sea disposal of contaminated dredged spoil and solid wastes is now a common practice. The sedimentation of particles is often accompanied by * Tel.: +852-2859-2622; fax: +852-2858-5415. E-mail address: [email protected] (C.-O. Ng). 0029-8018/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 0 0 ) 0 0 0 5 7 - 3
1394
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
Nomenclature a b C Cs D d E G g H h Im i k LM N n Pe Re Sc s T t UE UI UL Um US u uI u∗ Vm v wf x y z z1 z2 b g
surface wave amplitude interface wave amplitude chemical concentration sediment concentration dispersion coefficient mud depth eddy diffusivity of mass complex parameter given by Eq. (51) acceleration due to gravity complex parameter given by Eq. (43) water depth the imaginary part of imaginary unit wavenumber the Monin–Obukhov length cell function defined by Eqs. (24) and (25) local vertical coordinate pointing from the mud bottom Peclet number the real part of Schmidt number relative density of sediment solid wave period time depth-averaged Eulerian streaming velocity amplitude of uI depth-averaged mass transport velocity amplitude of um1 depth-averaged Stokes drift horizontal component of fluid velocity near-bottom velocity given by the potential wave theory shear velocity amplitude of vm1 vertical component of fluid velocity falling velocity horizontal coordinate vertical coordinate complex number (=1⫺i) complex parameter (=zf+z∗) complex parameter (=zf⫺z∗) complex parameter given by Eq. (58) ratio of fluid density (=rw/rm)
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
⌬ d ⑀ z h n x r s f ⌿ y 具·典 (·) (·)1 (·)2 (·)m (·)w (·)∗ (·˜)
1395
Stokes boundary layer thickness for mass diffusion Stokes boundary layer thickness for momentum diffusion wave steepness ratio of Stokes boundary layer thicknesses (=dm/dw) surface displacement von Karman’s constant eddy viscosity interface displacement density wave angular frequency ratio of Stokes boundary layer thicknesses due to momentum and mass diffusion (=dm/⌬m) complex function given by Eq. (49) complex parameter given by Eq. (50) depth averaging time averaging first-order quantity second-order quantity quantity of mud quantity of water complex conjugate normalized quantity
continuous phase exchange of chemicals between sorbed and dissolved forms. It is possible that sorption of dissolved toxic materials such as polychlorinated biphenyls (PCBs), pesticides and heavy metals by suspended sediment, followed by deposition of the sediment, can scavenge substantial amounts of toxic material from the water column. Deposited materials, on the other hand, may interact with benthic organisms to result in diverse products. For example, bacterial degradation of organic matter may lead to disaggregation of sediments, nutrient regeneration, and so on, while microorganisms mediate many chemical reactions which may alter toxic chemicals to an even more toxic form. Input of man-made wastes may also cause diagenesis of natural sediment. On top of these, sediments are constantly in a dynamic state of motion owing to resuspension by waves, transport by currents, pumping by tides, stirring by bioturbation and so on, until burial deep below the interface by other deposits. When predicting the fate and transport of a chemical, one is often confronted with the task of working out the rates of the various processes that can occur on a specific interface. Though a comprehensive study is much desirable, the scope of a model is frequently limited by practical constraints. Simplifying a model is possible, however, only when one is able to estimate a priori which processes are dominant given a set of parameters in a particular geophysical setting. In this connection, a good
1396
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
quantitative tool, conceptual or mathematical, is an essential element. This is especially true for sites like the benthic boundary layer for which field data are scarce. The aim of this work was to examine specifically the mass transport in the benthic boundary layer which is laden with a heavy suspended load and is oscillatory under surface waves. While emphasizing only the physical forcings, the present work will be of great practical value in yielding relations by which one can estimate the transport rates of benthic sorbed contaminants under the sole influence of waves, which are an important agent in sediment transport on continental shelves. Transport of littoral sediment due to the combined action of waves and currents has been extensively studied (e.g., see Fredsøe and Deigaard, 1992; Nielsen, 1992; Madsen, 1993, and references therein). Typically, wave action is responsible for keeping sediments in suspension, while currents provide the net transport. Theoretical models of sediment transport where waves play a direct role in both aspects of entrainment and net transport have been presented by Mei and co-workers (Mei and Chian, 1994; Mei et al. 1997, 1998). They obtained effective advection–dispersion equations for transport of a finite cloud of low-concentration fine suspended sediments under the pure action of surface waves. The advection is solely due to waveinduced steady streaming, while dispersion is caused by shear in the wave boundary layer. Both processes are effective at a time scale much longer than a wave period. In the above-mentioned dispersion models, the sediment concentration is assumed to be so low that the flow characteristics are essentially not affected by the particles. This assumption, however, breaks down when the solid concentration becomes a finite fraction of the fluid density. One direct effect of the presence of solid particles is to dampen the turbulent fluctuations, as can be inferred from the variation of von Karman’s constant with sediment concentration. The constant drops from its value of 0.4 in clear fluid to as low as 0.15 for a sufficiently heavy suspended load (Einstein and Chien, 1954). Einstein and Chien (1955) further advanced a modified velocity distribution equation for steady sediment-laden stream flows. On fitting with experimental data, they found it necessary to divide the flow into two zones. Close to the bed is a shallow layer (thickness being of the order of one-tenth of the total depth) of a heavy-fluid zone where the sediment is heavily concentrated. Overlying this zone is a light-fluid zone where the sediment concentration is small and does not change the fluid density. The coarser the particles, the larger the fraction of the solid population residing in the heavy-fluid zone. Therefore, in a concentrated sedimentladen flow, the heavy-fluid zone, because of its much larger solid–water ratio, will be responsible for transportation of most of the sorbed chemicals. The intention of the present work was to develop analytically a transport model for contaminated particles, which may originate from human dumping, in such a heavily loaded boundary layer under the pure action of progressive waves. The theory of Mei and Chian (1994) is extended to a two-layer wave boundary layer, based on the two-fluid-zone structure of Einstein and Chien (1955). We study a simplified problem in order to capture the essence of the physics. One of the idealizations is to model the abrupt yet continuous stratification by two distinct layers of homogeneous fluid: the upper fluid is clear water while the lower fluid is a denser liquid
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1397
suspension. By this construction, a sharp interface between the two fluids is assumed to persist. This will not be too strong an assumption in practice as long as the stratification is stable, as should be in the present case. If the bottom layer thickness is comparable to the Monin–Obukhov length, the layer will be well-mixed because the turbulence will prevail over buoyancy forces within a layer of this scale. Consequently vertical diffusion of mass may occur within the layer, but diffusion across the density interface that forms at the top of the stable layer is strongly inhibited. Momentum transport, on the other hand, can be transmitted across the density interface by internal waves. It has been reported that some high-density fluid mud suspensions appear to be stable, even under large shearing stresses (McCave, 1974). Our problem is described further in Section 2. Mass transport of a chemical takes place in the denser fluid of a two-layer wave boundary layer (also known as the Stokes boundary layer) under small-amplitude surface progressive waves. The basic governing equations and the assumptions regarding the scalings are first stated in Section 2 before variables are expanded in terms of the small wave steepness ⑀. This is followed by a perturbation analysis in Section 3. The analysis is based on the method of homogenization, which has been utilized to obtain effective macroscale equations for a wide range of flow and transport problems (Mei et al., 1996). In this case, effective transport equations are deduced up to O(⑀2), in which the advection velocity and dispersion coefficient show up as functions of fluid properties and flow characteristics. It will be shown that the advection velocity is equal to the depthaveraged mass transport velocity or Lagrangian drift, while the dispersion coefficient is always positive. Explicit expressions for the advection velocity and the dispersion coefficient are then deduced in Sections 4 and 5 respectively upon invoking the results of Ng (2000) and solving a canonical “cell” problem. Physical discussions are then presented in Section 6 to illustrate the dependence on various parameters of these transport coefficients, in dimensional and normalized forms.
2. Formulation As shown in Fig. 1, a train of two-dimensional, small-amplitude progressive waves is propagating on the surface of a clear water body of uniform depth h, which is overlying a thin, heavy-fluid layer of thickness d above the bottom. The bottom is assumed to be flat, impermeable and rigid. In other words, effects due to bedform and exchange of particles with the bed are not taken into consideration. The heavy fluid, which is essentially a well-mixed suspension, will also be referred to as “mud” for simplicity. The flows are turbulent. Like Mei and Chian (1994), we assume that near the bottom the eddy viscosities and diffusivities are constants in order to facilitate analytical development. Such an approximation is commonly employed in oceanographical studies and is believed to be good enough in many practical cases. Also, a sharp interface between the fluids is assumed, as discussed earlier. A twolayer Stokes boundary layer model that has recently been developed by Ng (2000) will be followed in this work. In an (x, y) coordinate system where x is the still water level and y is pointing
1398
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
Fig. 1.
Sketch of problem geometry.
vertically upward, the surface profile for progressive waves propagating in the positive x-direction is given by y⫽h(x, t)⫽Re{a ei(kx−st)},
(1)
where a is the wave amplitude, i is the imaginary unit, t is time, k is the wavenumber and s is the wave angular frequency. Without loss of generality, a and s are assumed to be real. The flow is essentially inviscid except very near the boundaries. At the leading order, k is also real and governed by the dispersion relation s2⫽gk tanh kh,
(2)
where g is the acceleration due to gravity. The near-bottom velocity given by the inviscid flow theory is uI⫽Re{UI ei(kx−st)}
sa where UI⫽ . sinh kh
(3)
The vertical extents of the bottom turbulent diffusion of momentum and mass are scaled by the corresponding Stokes boundary layer thicknesses: dm⬅(2nm/s)1/2 and ⌬m⬅(2Em/s)1/2,
(4)
where nm and Em are respectively the eddy diffusivity of momentum and mass. Typically the Schmidt number Sc⬅nm/Em is of order unity (supported by Reynolds analogy), and therefore the two Stokes boundary layer thicknesses are in general comparable to each other. We further assume that they are also of the same order of magnitude as the undisturbed mud depth d and the wave amplitude a, which are all much smaller than the wavelength 2p/k and the water depth h. The small wave steepness ⑀⬅ka苲kd苲kdm苲k⌬m1
(5)
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1399
will be used as an ordering parameter. For good mixing within the mud layer while maintaining a stable density interface on the top, the thickness of the layer d must also be in the order of the Monin– Obukhov length for suspended sediment stratification, which is given by LM⫽u3∗/(s⫺1)gwfCs,
(6)
where u∗ is the shear velocity, s is the specific weight of sediment solid, is von Karman’s constant, wf is the falling velocity of sediments and Cs is the volumetric concentration of sediments. To describe flow and transport near the bottom, it is more convenient to use a local vertical coordinate n⬅y+h+d, which points upward from the bottom of the mud layer (Fig. 1). The mud–water interface is given by n=d+x where x(x, t)⫽Re{b ei(kx−st)}
(7)
is the interface displacement with a complex amplitude b, whose modulus is an order of magnitude smaller than a. Let C be the concentration of a chemical in the mud (i.e., total mass of a chemical species, say in sorbed form, per unit volume of the heavy fluid) and (um, vm) be the horizontal and vertical components of the mud fluid velocity. Then the mass transport of the chemical is governed by
冉
冊
∂C ∂C ∂2C ∂2C ∂C ⫹⑀um ⫹⑀vm ⫽Em ⑀2 2 ⫹ 2 , ∂t ∂x ∂n ∂x ∂n
(8)
where the eddy diffusivity of mass is assumed to be the same in horizontal and vertical directions. Since mass exchanges with the bed and the overlying clear water are ignored, the normal flux at these interfaces must be zero: Em
∂C ⫽0 on n⫽0 ∂n
Em
冉
and
(9)
冊
∂C x 2 ∂3 C ∂2C ⫹O(⑀3)⫽0 on n⫽d. ⫹⑀x 2 ⫹⑀2 ∂n ∂n 2 ∂n3
(10)
The kinematic condition at the mud–water interface is ∂x ⫽v ⫹O(⑀) on n⫽d. ∂t m
(11)
Note that, in Eqs. (10) and (11), the conditions at the exact position n=d+x have been approximated by those at n=d using Taylor series expansions. Also, the small ordering parameter ⑀ has been inserted in Eqs. (8)–(10) to indicate the relative order of magnitude of the associated term, as previously shown by Mei and Chian (1994). These orders are based on the following scalings: x=O(k⫺1), n=O(d)=O(⌬m)=O(⑀k⫺1), t=O(s⫺1), um=O(UI), vm=O(⑀UI), x=O(⑀a) and the Peclet number Pe⬅UI/kEm=O(⑀⫺1). In the absence of steady currents at the leading order, the pri-
1400
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
mary time scale t2 over which wave-induced transport becomes effective is that due to spreading by dispersion over one horizontal length scale, which is two orders of magnitude longer than a wave period since t2/t=O(s/Emk2)=O(⑀⫺2). Let us now expand the variables and the time derivative: C→C0⫹⑀C1⫹⑀2C2⫹O(⑀3),
(12)
(um, vm)→(um1, vm1)⫹⑀(um2, vm2)⫹O(⑀2),
(13)
∂/∂t→∂/∂t⫹⑀ ∂/∂t2.
(14)
2
The leading order velocity components can be further expanded (Ng, 2000): (um1, vm1)⫽Re{(Um, Vm) ei(kx−st)},
(15)
where Um(n) and Vm(n) are related to each other by the continuity equation: ikUm⫽⫺V⬘m.
(16)
A relation for the amplitude of the interface displacement is also found by substituting Eqs. (7) and (15) into Eq. (11): b⫽is−1Vm(d).
(17)
Note that the second-order velocity um2 has a steady component, known as the Eulerian streaming velocity um2, which will also appear in later deductions.
3. Effective transport equations Successive orders of perturbation equations are obtainable by substituting Eqs. (12)–(14) into Eqs. (8)–(10), which are discussed as follows. The problem at O(1) is ∂2C0 ∂C0 ⫽Em 2 ∂t ∂n
in 0⬍n⬍d
(18)
and ∂C0 ⫽0 on n⫽0, d. ∂n
(19)
The leading order concentration C0 is taken to represent the smooth component whose variations with the short time t die out in the long term (Mei and Chian, 1994; Mei et al., 1996). Hence it is clear that, on ignoring the transience, C0 is independent of n as well, or C0⫽C0(x, t2).
(20)
On simplifying using Eq. (20) and substituting Eq. (15), the O(⑀) equation becomes
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1401
∂C0 ∂C1 ∂2C1 ⫹Re{Um ei(kx−st)} ⫽Em 2 in 0⬍n⬍d, ∂t ∂x ∂n
(21)
which is subject to the boundary conditions ∂C1 ⫽0 on n⫽0, d. ∂n
(22)
By linearity of Eq. (21), the following form for C1 is suggested: ∂C0 , C1(x, n, t)⫽Re{N(n) ei(kx−st)} ∂x
(23)
where the function N(n) is governed by Em
d2N ⫹isN⫽Um dn2
in 0⬍n⬍d
(24)
and dN ⫽0 on n⫽0, d. dn
(25)
Solutions of N(n) will be sought in Section 5 when we derive expressions for the dispersion coefficient. Let us now consider equations of O(⑀2). In a conservative form, the perturbation equation of this order reads
冉
∂ ∂C0 ∂2C0 ∂2C2 ∂C0 ∂C2 ∂ ⫹ ⫹ (um1C1)⫹ (vm1C1)⫹um2 ⫽Em ⫹ ∂t2 ∂t ∂x ∂n ∂x ∂x2 ∂n2
冊
in 0⬍n
(26)
⬍d, while the boundary conditions are ∂C2 ⫽0 on n⫽0 ∂n
(27)
∂C2 ∂2C1 ⫹x 2 ⫽0 on n⫽d. ∂n ∂n
(28)
and
Taking the depth average followed by the time average of Eq. (26) while using the boundary conditions yields
冋
1 ∂2C1 ∂C0 ∂ ⫹ 具um1C1典⫹ vm1C1⫹Emx 2 ∂t2 ∂x d ∂n
册
⫹具um2典 n=d
∂C0 ∂2C0 ⫽Em 2 , ∂x ∂x
(29)
where the angle brackets and the overbar denote the depth average and the time average over a wave period T, respectively:
1402
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
冕 d
冕
t⫹T
1 1 具f典⬅ f dn, ¯f⬅ d T 0
f dt.
(30)
t
On substituting Eqs. (15, 7) and (23), the second and third terms on the left-hand side of Eq. (29) can be developed as follows: 1 ∂2C0 ∂ 具um1C1典⫽ Re具UmN∗典 2 , ∂x 2 ∂x
(31)
where the asterisk denotes the complex conjugate, and
冋
册 冋 冉
∂2C1 1 vm1C1⫹Emx 2 d ∂n ⫽
冋
册
1 d2N∗ ⫽ Re VmN∗⫹Emb 2 dn n=d 2d
冊册
d2N∗ 1 Re is−1Vm Em 2 ⫺isN∗ 2d dn
∂C0 n=d ∂x
(32)
∂C0 ∂C0 1 ⫽ Re[is−1Vm(d)U∗m(d)] , ∂x 2d ∂x n=d
where Eqs. (17) and (24) have been used in Eq. (32). Putting these back to Eq. (29), we finally have the effective transport equation: ∂2C0 ∂C0 ∂C0 ⫹UL ⫽[Em⫹D] 2 , ∂t2 ∂x ∂x
(33)
where UL⫽UE⫹US,
(34)
in which UE⫽具um2典
(35)
is the depth-averaged Eulerian streaming velocity, US⫽
1 Re[is−1Vm(d)U∗m(d)] 2d
(36)
is an additional component of the advection velocity, and 1 D⫽⫺ Re具UmN∗典 2
(37)
is the dispersion coefficient. Eq. (33) governs the transport of a chemical species in the heavy-fluid layer, in which the wave action alone induces both advection and dispersion, which become effective over a long time scale T2=O(⑀⫺2T). Some further analytical relationships can be developed as follows. We first show that the additional advection velocity component US is actually equal to the depth-averaged Stokes drift. For the present two-dimensional problem, Stokes drift (Longuet-Higgins, 1953) is given by:
冉冕 冊 冉冕 冊 um1 dt
∂um1 ⫹ ∂x
vm1 dt
∂um1 ⫽Re{is−1Um ei(kx−st)}Re{⫺V⬘m ei(kx−st)} ∂n
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1 ∗ ⬘ ⫹Re{is−1Vm ei(kx−st)}Re{U⬘m ei(kx−st)}⫽ Re{is−1(Um Vm⫹VmU∗⬘ m )} 2
1403
(38)
1 ⫽ Re{is−1(VmU∗m)⬘}, 2 whose depth average is clearly equal to US: 1 1 ∗ ∗ Re具is−1(VmUm )⬘典⫽ Re[is−1Vm(d)Um (d)]⫽US. 2 2d
(39)
Hence in Eq. (33) the advection velocity, composed of Eulerian streaming velocity and Stokes drift, amounts to the depth-averaged Lagrangian drift or mass transport velocity of individual fluid particles. We next show that D is always positive, which is a necessary condition for a physically admissible diffusion coefficient. On substituting Eq. (24) for Um into Eq. (37), we may obtain in general
再
Em Em 1 D⫽⫺ Re具(EmN⬙⫹isN)N∗典⫽⫺ Re具N∗N⬙典⫽⫺ Re [N∗N⬘]d0 2 2 2d
冕 d
(40)
冎 冕 d
Em 兩N⬘兩2 dn⬎0, ⫺ N⬘N⬘∗ dn ⫽ 2d 0
0
where integration by parts and the boundary conditions (25) have been used.
4. Advection velocity Let us now derive explicit expressions for the mass transport velocity. From Ng (2000), the following relations may be invoked: Um(n˜ )⫽(g⫺g cosh zn˜ ⫹H sinh zn˜ )UI
(41)
z Vm(n˜ )⫽ kdm[g(zn˜ ⫺sinh zn˜ )⫹H(cosh zn˜ ⫺1)]UI, 2
(42)
and
where H⫽
g(1−g)+g2 cosh zd˜ +gz sinh zd˜ . z cosh zd˜ +g sinh zd˜
(43)
In Eqs. (41) and (42), the following definitions of notation have been used: z⬅1⫺i is a complex constant,
(44)
1404
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
(n˜ , d˜ )⬅(n, d)/dm
(45)
are the normalized vertical coordinate and mud depth with respect to the momentum Stokes boundary layer thickness, g⬅rw/rm
(46)
is the ratio of the density of water (rw) to that of mud (rm), and z⬅dm/dw⫽(nm/nw)1/2
(47)
is the ratio of the momentum Stokes boundary layer thickness in mud (dm) to that in water (dw), where nw is the eddy viscosity of water. The Eulerian streaming velocity has also been found by Ng (2000): um2(n˜ )⫽ks−1U2IRe{⌿(n˜ )⫹yn˜ ⫹3g2/4},
(48)
where ⌿(n˜ )⫽(1/8)(g2⫹兩H兩2) cosh 2n˜ ⫹(1/8)(g2⫺兩H兩2) cos 2n˜ ⫺(g/8)(H ⫹H∗) sinh 2n˜ ⫹(ig/8)(H⫺H∗) sin 2n˜ ⫺(z∗gH∗/2)n˜ cosh z∗n˜ ⫺(g2
(49)
⫺i兩H兩 /2) cosh z n˜ ⫹(z g /2)n˜ sinh z n˜ ⫹(gH ⫺igH/2) sinh z n˜ 2
∗
∗ 2
∗
∗
∗
and y⫽⫺(1/4)(g2⫹兩H兩2) sinh 2d˜ ⫹(1/4)(g2⫺兩H兩2) sin 2d˜ ⫹(g/4)(H ⫹H∗) cosh 2d˜ ⫺(ig/4)(H⫺H∗) cos 2d˜ ⫹(z∗g2/2⫹igH∗d˜ ⫹z兩H兩2/2) sinh z∗d˜ ⫺(z∗gH∗/2⫹ig2d˜ ⫹zgH/2) cosh z∗d˜ ⫺(z∗gG∗/2z)[z∗zgd˜ ⫺igz sinh zd˜
(50)
⫹izH(cosh zd˜ ⫺1)⫹1⫹iG]⫺g兩G兩 /2z⫺(z/2)[g(zd˜ ⫺sinh zd˜ )⫹H(cosh zd˜ ⫺1)][g cosh z∗d˜ ⫺H∗ sinh z∗d˜ ⫹gG∗], 2
in which G⫽
−gz−(1−g)z cosh zd˜ . z cosh zd˜ +g sinh zd˜
(51)
Also, the depth average of ⌿ is 具⌿典⫽(1/16d˜ )(g2⫹兩H兩2) sinh 2d˜ ⫹(1/16d˜ )(g2⫺兩H兩2) sin 2d˜ ⫺(g/16d˜ )(H ⫹H∗)(cosh 2d˜ ⫺1)⫺(ig/16d˜ )(H⫺H∗)(cos 2d˜ ⫺1)⫹(g2/2⫹3zgH∗/4d˜ ⫺z∗gH/4d˜ ) cosh z∗d˜ ⫺(gH∗/2⫹3zg2/4d˜ ⫺z∗兩H兩2/4d˜ ) sinh z∗d˜ ⫺3zgH∗/4d˜
(52)
⫹z∗gH/4d˜ . Hence, the depth-averaged Eulerian streaming velocity is given by UE⫽具um2典⫽ks−1U2IRe{具⌿典⫹yd˜ /2⫹3g2/4},
(53)
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1405
while the depth-averaged Stokes drift is, from Eq. (36), kU2I Re{(zd˜ )−1[g(zd˜ ⫺sinh zd˜ )⫹H(cosh zd˜ ⫺1)][g⫺g cosh z∗d˜ 2s ⫹H∗ sinh z∗d˜ ]}.
US⫽
(54)
5. Dispersion coefficient To obtain explicit expressions for the dispersion coefficient, we first need to solve for N from Eq. (24), which after substituting Eq. (41) can be written as UId2m d 2N ⫺(zf)2N⫽(g⫺g cosh zn˜ ⫹H sinh zn˜ ) , 2 dn˜ Em
(55)
where f⫽dm/⌬m⫽(nm/Em)1/2⫽Sc1/2
(56)
is the ratio of the Stokes boundary layer thicknesses due to momentum and mass diffusion, and is equal to the square root of the Schmidt number. As discussed earlier, this ratio should be of order unity. We however assume that f⫽1 for generality and may obtain the following solution satisfying Eqs. (55) and (25) id2mUI [⫺H sinh zfn˜ ⫹b cosh zfn˜ ⫹fH sinh zn˜ ⫺fg cosh zn˜ N(n˜ )⫽ 2f(1−f2)Em
(57)
⫺g(1⫺f2)/f], where b⫽
H(cosh zfd˜ −cosh zd˜ )+g sinh zd˜ . sinh zfd˜
(58)
We may now plug Eqs. (41) and (57) into Eq. (37), and obtain after some algebra: D⫽
再
(gH+bH∗)[cosh z1d˜ −1]−(gb+兩H兩2) sinh z1d˜ fs−1U2I Im 4(1−f2)d˜ z1
(59)
冎
(gH−bH∗)[cosh z2d˜ −1]−(gb−兩H兩2) sinh z2d˜ , ⫹ z2 where z1⫽zf⫹z∗,
z2⫽zf⫺z∗.
(60)
Using L’Hospital’s rule, one may show that the dispersion coefficient tends to the following finite limit when f approaches unity:
1406
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
s−1U2I {Re([g2 sinh z∗d˜ ⫹gH∗(cosh zd˜ ⫺cosh z∗d˜ )]/sinh zd˜ ) lim D⫽ 8 f→1 ⫹(1/4d˜ )(g2⫹兩H兩2) sinh 2d˜ ⫺(1/4d˜ )(g2⫺兩H兩2) sin 2d˜ ⫺(gHr/2d˜ )(cosh 2d˜ ⫺1) ⫺(gHi/2d˜ )(cos 2d˜ ⫺1)⫺兩H兩2},
(61)
where Hr and Hi are respectively the real and imaginary parts of H.
6. Physical discussions Explicit expressions have now been obtained for the effective advection velocity UL=UE+US, and the dispersion coefficient D. These are functions of the wave frequency s, near-bottom velocity UI (which depends on the wave amplitude, wavenumber k and water depth h), and the following ratios: g=rw/rm, z=(nm/nw)1/2, f=(nm/Em)1/2 and d˜ =d/dm. Clearly, a heavy-fluid suspension is denser than water, so the density ratio must be less than unity: g⬍1. Also, zⱕO(1) because an eddy viscosity for wall turbulence typically increases with the distance from the wall, and also the mixing length for momentum in clear water should be greater than that in a particle-laden liquid. In general, the parameter z controls the relative rates of momentum diffusion across the mud and the water boundary layer. The ratio f, or the square root of the Schmidt number, is equal to unity by Reynolds analogy. However, this number can be different from unity since the mixing length for mass, because of inertia, is somewhat shorter than that for momentum; but, on the other hand, solid particles may be thrown further to the outside of eddies by centrifugal forces (Vanoni, 1975). Some sample values of the transport coefficients have been computed and are shown in Table 1 for the cases h=20 and 40 m and T=5, 10, 15 and 20 s, where a=0.5 m, g=0.75, z=0.5, f=1.0 and d˜ =2.0. The wave periods are that typical of wind waves. It is clear that the velocities and the dispersion coefficient increase with the near-bottom velocity UI, which in turn is greater for longer waves over a shallower water layer. Except for rather short waves, the mass transport velocity is in the order of 1 cm/s (Eulerian streaming velocity and Stokes drift being comparable to each other), while the dispersion coefficient is the range of O(1)–O(100) cm2/s. These values are at least of the same order of magnitude as that due to tidal currents, and confirm the importance of the action of wind waves on the transport of matter in the benthic boundary layer. Of course, for a deep water layer or large kh, these waveinduced transport mechanisms can become subdominant. The variations of the transport coefficients, in dimensionless form, with the independent parameters are further examined in Figs. 2–5. Non-dimensional coefficients are defined as follows: ˜ S, U ˜ L)⬅(UE, US, UL)/ks−1UI, D ˜ ⫽D/s−1U2I. ˜ E, U (U
(62)
˜ L (solid lines) and U ˜ E (dashed lines) as functions of d˜ for various g Fig. 2 shows U and z. The value of Stokes drift can be obtained from the difference of the corre-
0.161 3.23 0.050 0.18 0.18 0.36 0.35
5
T (s)
k (m⫺1) kh UI (m/s) UE (cm/s) US (cm/s) UL (cm/s) D (cm2/s)
20
h (m)
0.052 1.04 0.255 0.60 0.57 1.17 18.1
10 0.032 0.64 0.308 0.67 0.64 1.31 39.7
15 0.023 0.46 0.327 0.69 0.66 1.35 59.5
20
5 0.161 6.44 0.002 0.007 0.007 0.014 0.0006
40
0.043 1.72 0.117 0.23 0.22 0.45 3.79
10
0.024 0.96 0.188 0.31 0.30 0.60 14.7
15
0.017 0.68 0.214 0.33 0.32 0.65 25.6
20
Table 1 Sample model values of velocities and dispersion coefficient under typical wind-wave conditions, for a=0.5 m, g=0.75, z=0.5, f=1.0 and d˜ =2.0
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411 1407
1408
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
˜ L (solid lines) and Eulerian streaming veloFig. 2. Normalized depth-averaged mass transport velocity U ˜ E (dashed lines) as functions of d˜ and g, for (a) z=0.1 and (b) z=1.0. city U
Fig. 3.
˜ as function of f and d˜ , for g=z=0.5. Normalized dispersion coefficient D
sponding pair of curves. Clearly the advection velocity components increase with d˜ and g, or the advection rate is higher for a thicker layer of lighter mud. This is reasonable because for such a mud layer more fluid elements can be induced into more appreciable motion by the waves. The motion of denser mud tends to be more sluggish under wave action (Ng, 2000). An increase in z will however lower the velocities, especially for smaller d˜ and g. This is so because a larger z for a fixed nm means a smaller nw or a smaller rate of momentum diffusion across the water boundary layer. ˜ is shown in Fig. 3, where g=z=0.5 are assumed. The The effect of f=Sc1/2 on D dispersion coefficient reaches a maximum at a value of f which is of the order of unity, the specific values being dependent on the mud layer thickness d˜ . For instance,
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1409
Fig. 4.
˜ as function of d˜ and g, for z=0.1 and (a) f=1.0 and (b) f=2.0. Normalized dispersion coefficient D
Fig. 5.
˜ as function of d˜ and g, for z=1.0 and (a) f=1.0 and (b) f=2.0. Normalized dispersion coefficient D
˜ is the maximum at f=1.1 when d˜ =2, but at f=2.2 when d˜ =1. The assumption of D Reynolds analogy or Sc=1 will in general lead to a dispersion coefficient that is closer to the peak value for a thicker mud layer. ˜ increases with d˜ until a maximum is reached, Figs. 4 and 5 further show that D ˜ and the corresponding d˜ being functions of the parathe values of the maximum D ˜ occurs when meters g, z and f. For the cases shown in the plots, the maximum D d˜ is in the range 1–2.5. Also, a lower g or a denser mud layer will correspond to a smaller dispersion coefficient, the effect being more prominent for a larger z. Increas˜ to occur at a smaller d˜ . In general, one ing the value of f will cause the peak D may expect that the dispersion coefficient is near its peak value when the mud depth is one to two times the Stokes boundary layer thickness. The dispersion coefficient will drop significantly as the mud layer becomes too thin or too thick as compared with the Stokes boundary layer thickness.
1410
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
7. Concluding remarks In this work, we have presented an analytical model for transport of a chemical species sorbed on to suspended sediments in a heavily loaded benthic boundary layer under the pure action of surface progressive waves. The key assumptions are: (1) the stratified boundary layer is modelled by a two-layer boundary layer with a sharp interface; (2) the heavy-fluid layer has a thickness comparable to the Monin–Obukhov length, the Stokes boundary layer thicknesses and the wave amplitude; (3) fluid properties including eddy viscosity and diffusivity are constant within a fluid layer; and (4) bedform and exchange with the bed are ignored. Following an asymptotic method of averaging which is based on the homogenization technique, we have deduced an effective transport equation, as formally given by Eqs. (33)–(37). The effective advection velocity is found to be equal to the depth average of the mass transport velocity, comprising Eulerian streaming velocity and Stokes drift. Also, the dispersion coefficient can be formally shown to be positive definite. Explicit expressions for these transport coefficients are given in Eqs. (53, 54) and (59) as functions of characteristics of the waves and the boundary layer. We have also shown that the benthic advection and dispersion rates due to typical wind waves alone can be as significant as those due to tidal currents. The dispersion coefficient is found to reach a maximum when the depth of the heavy fluid is slightly thicker than the Stokes boundary layer thickness, and when the Schmidt number is slightly larger than unity. Despite the simplifying assumptions, the present theory provides one with an analytical tool to estimate the effects of pure waves on the transport of a benthic species. Of course, other factors that have not been taken into consideration may be just as influential as the waves. A more complete model may have the individual components due to various factors lumped into the advection and dispersion coefficients. Our theory can offer a modular component due to pure waves. Future efforts to advance a more comprehensive transport model are deemed worthwhile.
Acknowledgements The work described in this paper was jointly supported by two grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Projects HKU 7117/99E and NSFC/HKU 8).
References Einstein, H.A., Chien, N., 1954. Second approximation to the solution of the suspended-load theory. Report No. 3. University of California, Institute of Engineering Research. Einstein, H.A., Chien, N., 1955. Effects of heavy sediment concentration near the bed on velocity and sediment distribution. Report No. 8. University of California, Institute of Engineering Research. Fredsøe, J., Deigaard, R., 1992. Mechanics of Coastal Sediment Transport. World Scientific, Singapore. Longuet-Higgins, M.S., 1953. Mass transport in water waves. Phil. Trans. R. Soc. Lond. A245, 535–581.
C.-O. Ng / Ocean Engineering 28 (2001) 1393–1411
1411
Madsen, O.S., 1993. Sediment Transport on the Shelf. Parsons Lab., MIT, Cambridge, MA. McCave, I.N. (Ed.), 1974. The Benthic Boundary Layer. Plenum Press, New York. Mei, C.C., Chian, C., 1994. Dispersion of small suspended particles in a wave boundary layer. J. Phys. Oceanogr. 24, 2479–2495. Mei, C.C., Auriault, J.L., Ng, C.O., 1996. Some applications of the homogenization theory. In:. Hutchinson, J.W., Wu, T.Y. (Eds.), Advances in Applied Mechanics, vol. 32. Academic Press, California, pp. 277–348. Mei, C.C., Fan, S.J., Jin, K.R., 1997. Resuspension and transport of fine sediments by waves. J. Geophys. Res. 102 (C7), 15807–15821. Mei, C.C., Chian, C., Ye, F., 1998. Transport and resuspension of fine particles in a tidal boundary layer near a small peninsular. J. Phys. Oceanogr. 28 (11), 2313–2331. Ng, C.O., 2000. Water waves over a muddy bed: a two-layer Stokes’ boundary layer model. Coastal Eng. 40 (3), 221–242. Nielsen, P., 1992. Coastal Bottom Boundary Layers and Sediment Transport. World Scientific, Singapore. Vanoni, V.A., 1975. Sedimentation Engineering. American Society of Civil Engineering, New York.