A numerical investigation of lutocline dynamics and saturation of fine sediment in the oscillatory boundary layer

Authors


Abstract

[1] Understanding the state of the muddy seabed is critical to sediment transport, hydrodynamic dissipation and seabed properties. However, this endeavor is challenging because the availability and settling velocity of sediment in muddy environments are highly variable. For a given Reynolds number, typical of fine sediment settling in a moderately energetic muddy shelf, recent 3D numerical simulations have revealed four distinct regimes of wave-induced fine sediment transport. These regimes depend on the availability (or concentration) of sediment and range from well-mixed condition to the formation of lutocline, and eventually a complete flow laminarization. By keeping the sediment availability unchanged, this study further demonstrates the existence of these flow regimes for a range of sediment settling velocities. Simulation results suggest that when settling velocity is larger, the location of the lutocline becomes lower (closer to the bed) and the flow eventually laminarizes when there is further increase in the settling velocity. Hence, the dynamics of lutocline is clearly related to the transition between these flow regimes. The vertical flow structure in the presence of lutocline is revealed through the budgets of sediment flux and turbulent kinetic energy. The suppressed mixing in the lutocline layer is further illustrated from a new perspective, i.e., the budgets of turbulent suspension and concentration fluctuation variance. The concept of saturation, commonly used for tidal boundary layer, is extended here for wave boundary layer.

1. Introduction

[2] Fluid mud is ubiquitous in many estuaries and continental shelves due to abundant riverine supply of fine sediment [Wolanski et al., 1988; Kineke et al., 1996; Traykovski et al., 2000]. One of the most intriguing features of such fine-grained sediment transport is that when sufficient amount of fine sediment is available, the suspension does not follow a well-mixed profile, as would be predicted by the classic Rouse profile [Rouse, 1937] due to its small settling velocity. Instead, significant amount of suspended sediment is confined near the bed, forming a pronounced sharp concentration gradient, called lutocline, due to the attenuation of carrier flow turbulence by the suspended sediment [e.g., Noh and Fernando, 1991; Trowbridge and Kineke, 1994; Huppert et al., 1995; Winterwerp, 2001, 2006]. A lutocline is the front between the two layers of comparatively high and low sediment concentration. The two-way coupled turbulence-sediment interactions and the resulting formation of lutocline give several unique features of fine sediment transport that have remarkable effects on the fate of the fine sediment, coastal morphology, hydrodynamic dissipation, underwater acoustic wave propagation, and exchange of substances (e.g., gas, nutrient and pollutant) with the seabed.

[3] For example, the formation of a high sediment concentration layer below the lutocline allows large density anomaly to be created, together with the influence of downslope gravitational effect, may yield significant offshore fine sediment transport on the continental shelf despite the shelf slope is often very mild [Traykovski et al., 2000, 2007; Wright et al., 2001; Wright and Friedrichs, 2006; Hsu et al., 2009]. The formation of the so-called wave-supported gravity-driven mudflows not only provides a key offshore transport mechanism of sediment source to sink, it also determines the unique existence of clinoform of muddy coast morphology [Friedrichs and Wright, 2004]. Furthermore, the formation of fluid mud may cause significant damping of surface wave energy [e.g., Sheremet and Stone, 2003; Elgar and Raubenheimer, 2008] through its highly viscous and/or non-Newtonian rheological stresses [e.g., Dalrymple and Liu, 1978]. The dynamics of fluid mud and the lutocline formation are also important for many engineering applications such as rapid siltation in the waterway [e.g., Wolanski et al., 1992].

[4] In the literature, fluid mud is a topic of interest in the context of tide-dominant estuary and river mouth [Smith and Kirby, 1989; Wolanski et al., 1988; Kineke et al., 1996; McAnally et al., 2007], wave-driven continental shelves [Traykovski et al., 2000, 2007] and even particulate jet flows due to aquifer water release to a lake [Colomer et al., 2002]. Noted from these examples, fluid mud processes can take place at vastly different spatial and temporal scales. For example, in a wave driven fluid motion the length scale of motion is on the order of centimeters whereas the length scale of tidal-driven fluid mud can be of several meters [Wolanski et al., 1988; Kineke et al., 1996]. Since the formation of lutocline is a process of dynamic equilibrium between turbulence production and buoyancy flux, i.e., a critical Richardson number [Noh and Fernando, 1991; Huppert et al., 1995], its evolution and mixing characteristics are expected to be quite different between the thick-and-gradual tidal boundary layer and the thin-and-rapid wave boundary layer [e.g., Winterwerp, 2001; Cantero et al., 2009; Ozdemir et al., 2010a].

[5] Turbulence modulation due to the presence of sediment has long been a topic of interest in the context of fluvial sediment transport [e.g., Vanoni, 1946; Bagnold, 1956; Lyn, 1988]and the concept of saturation has been suggested in the studies on turbidity current [Parker et al., 1986] and tidal-driven fluid mud transport [Winterwerp, 2001, 2006]. Motivated by field observations on wave-supported gravity-driven mudflows [Traykovski et al., 2000; Wright et al., 2001], several recent studies focus on wave-driven fluid mud transport and the related wave-current boundary layer processes based on Reynolds-averaged approaches [Harris et al., 2004; Traykovski et al., 2007; Hsu et al., 2007, 2009]. There are several challenges in modeling fluid mud transport, especially in the context of wave boundary layer. First, it is not completely clear what is the minimum set of governing equations and turbulence closure for the two-way coupled particle-laden flow system because turbulence modulation due to the presence of particles can be caused by sediment-induced density stratification and inertia effects [Balachandar and Eaton, 2010]. Second, a typical continental shelf with significant mud presence is not very energetic. A simple estimate for wave boundary layer at Eel shelf gives a Stokes Reynolds number ReΔ no more than 1000 (using wave velocity amplitude of 0.56 m/s and wave period of 10 s), where the wave boundary layer is not fully turbulent but in fact is in the intermittently turbulent regime [Hino et al., 1983]. Hence, to accurately compute transitional turbulent flow, all the scales of turbulence should be resolved and a Direct Numerical Simulation (DNS) approach is needed.

[6] There are clearly many complicated mechanisms inherently existent in fluid mud transport, such as flocculation, non-Newtonian behaviors, hindered settling, and turbulence modulation [e.g., McAnally et al., 2007]. We believe an evaluation of the interaction between flow turbulence and fine particles is a necessary step before we can incorporate other complicated mechanisms for a realistic fluid mud transport simulation. In this study, it is our objective to investigate the interaction between the fine sediment and the flow turbulence in order to form a benchmark toward future modeling of more realistic fluid mud transport.

[7] Assuming fine particles of very small particle response time and dilute flow, the equilibrium Eulerian approximation [Ferry and Balachandar, 2001] has been applied to the two-phase formulation to study turbidity currents [Cantero et al., 2008]. Ozdemir et al. [2010a] recently adopted this formulation for fine sediment transport in the wave boundary layer. The resulting equations incorporate the back effect of sediment on carrier flow turbulence only due to sediment induced density stratification. Using a highly accurate pseudo-spectral scheme and resolving all the scales of turbulence, they investigated turbulence-sediment interaction at a Stokes Reynolds number, ReΔ, of 1000, typical of an energetic muddy shelf. By varying the amount of available sediment concentration they observed the existence of four flow regimes of wave-induced fine sediment transport for a given sediment settling velocity of ∼0.5 mm/s: 1) a well-mixed sediment concentration with no modulation of carrier flow turbulence in case of very dilute particle concentration, 2) formation of a sharp sediment concentration gradient in the water column, i.e., lutocline, at about a near bed sediment concentration of O(1 ∼ 10) kg/m3, and flow below the lutocline remains turbulent, 3) nearly laminar concentration profile of both the sediment and fluid phases but marked by episodic burst events due to shear instability arising during flow reversal. This regime is observed at a near bed concentration of several tens of kg/m3, 4) at concentrations of O(100) kg/m3 or greater, a complete laminarization throughout the wave cycle is observed. The existence of these flow regimes has critical implications to our capability toward assessing the state of the muddy seabed and to further understand various applications related to fine sediment transport discussed previously. For instance, understanding the transition between regime 1 and regime 2, and further quantifying the evolution of lutocline in regime 2 will allow us to estimate the amount of offshore fine sediment transport [Harris et al., 2004]. Moreover, if a criterion exists for the transition between regime 2 and regime 3 (and 4), i.e., a saturation condition for wave-induced fine sediment, then many practical modeling of fine sediment transport can become rather simple because laminar solutions can be applied to regimes 3 and 4.

[8] While Ozdemir et al. [2010a] identified these flow regimes for different sediment availability; their investigation lacks the information on the effects of settling velocity which is a critical parameter of considerable variation in reality. It is very likely that settling velocity has a direct impact on the flow regimes as it strongly influences the local concentration distribution, and hence modifies the turbulence. Therefore, investigating the effect of settling velocity on flow turbulence and evaluating the existence of these flow regimes are the main objectives of this study. Moreover, the fact that regime 2 exists between regime 1 where no turbulence modulation is observed and regime 3 where turbulence is nearly collapsed, makes regime 2 very critical and deserves further investigation.

[9] The structure of this paper is as follows. The motivation and a brief literature review have been presented in section 1. The numerical methodology with its underlying assumptions and the list of simulations to be investigated are given in section 2. Simulation results and findings are given in section 3. Related discussions on the results are given in section 4. Finally, concluding remarks are given section 5.

2. Governing Equations and Numerical Methodology

[10] In this study, a simplified Eulerian-Eulerian two-phase flow model for fine sediment transport is adopted. This method is developed for fine particulate flow simulations where the particle Stokes number, St, defined as the ratio of particle response time (see equation (8)) to the characteristic timescale of the turbulent flow, is sufficiently smaller than unity [Balachandar and Eaton, 2010]. When this constraint is satisfied, the particle phase velocity can be represented as a vectorial sum of fluid velocity, settling velocity of the sediment and an expansion of the Stokes number, St. Algebraic approximation of particle phase velocity eliminates the necessity of solving particle phase momentum equations and hence the resulting numerical algorithm be greatly simplified. The accuracy of the model has been tested extensively in gravity flow and turbidity current simulations [Cantero et al., 2008, 2009].

[11] We consider fine sediment transport in an oscillatory channel that is statistically fully developed in the streamwise (x) and spanwise (z) directions (see Figure 1). Gravitational acceleration is defined to point toward the negative y-direction. The governing equations are solved in a computational box of rectangular prism that is sufficient to capture the largest turbulent eddies. The oscillatory channel is driven by a prescribed time series of free-stream velocity to mimic the wave forcing:

equation image

where the free-stream velocity amplitude is represented by equation image0. The angular frequency is given by equation image = 2π/equation image, with wave period, equation image. The tilde cap indicates that the variables are dimensional. In the implementation, the wave forcing shown in equation (1) corresponds to a time dependent mean streamwise pressure gradient in the momentum equation:

equation image

in which ρf is the fluid density. The Stokes' boundary layer thickness is defined as

equation image

with vf being the kinematic viscosity of the carrier fluid (water). A wave period of equation image = 10 s, and equation image0 = 0.56 m/s are selected to characterize the most energetic continental shelf conditions where fine sediments are present in the form of high concentration mud suspension or fluid mud [Traykovski et al., 2000]. Hence, the Stokes' boundary layer thickness is calculated to be 0.18 mm and Stokes' Reynolds number, which is given as follows:

equation image

becomes ReΔ = 1000. Though we have selected the Stokes' boundary layer thickness as the length scale, in most coastal research, orbital semi-excursion length, equation image = equation image0/equation image is used [see, e.g., Jensen et al., 1989]. The relationship between the Stokes' boundary layer thickness and the orbital semi-excursion length can be given as follows:

equation image
Figure 1.

Descriptive sketch of the computational domain used in all the simulations. Boundary conditions for fluid phase velocity are given as no-slip wall boundary conditions (NSWBC) at the top and the bottom wall and periodic boundary conditions (PBC) at the end plane of the computational domain. For sediment concentration, PBC are used at the same planes as fluid phase. At the top and the bottom wall the boundary condition is given by equation (14).

[12] Therefore, the wave orbital Reynolds number, i.e.,

equation image

becomes Rea = 500,000. From this point forward, all governing equations are given in non-dimensional form without the tilde cap.

[13] The major approximation of the present simplified two-phase flow formulation is that particles are assumed to be of small response time. The simulations conducted here consider the largest sediment (silt) diameter to be equation image = 42 μm, which corresponds to a settling velocity of 1.66 mm/s calculated based on the Stokes' settling law:

equation image

in which g = 9.8 m/s2 is the gravitational acceleration and s is the specific gravity of sediment, specified as 2.65. It should be mentioned that the maximum volumetric sediment concentration calculated in this study is no more than 5%, which is observed only very close to the bed when the flow is laminarized (see section 3.1 for details). Therefore, the hindered settling effect is neglected for simplicity.

[14] The particle response time is defined as

equation image

[15] The largest particle response time studied here is only equation imagep = 2.6 × 10−4 s. We nondimensionalize all the physical quantities using the free-stream velocity amplitude equation image0 as the velocity scale and the Stokes boundary layer thickness, equation image, as the length scale. Hence, the nondimensionalized particle response time, i.e., the Stokes number, St = equation imagepequation image0/equation image, is only 0.081 which allows us to utilize the equilibrium approximation [Balachandar and Eaton, 2010] and to further ignore the inertial terms due to carrier fluid motion. In this study, the dimensionless particle velocity Up is written as the vectorial sum of the non-dimensional fluid velocity Uf and the settling velocity of the particles, Vs:

equation image

where e2 is the unit vector in the y direction (direction of gravitational field). As discussed previously, with the dilute flow assumption, we further adopt Boussinesq approximation and neglect the sediment rheological stresses [Ozdemir et al., 2010a]. Hence, the fluid phase velocity satisfies the continuity equation:

equation image

and the fluid phase momentum equation can be given as

equation image

[16] In equation (11), e1 and e2 are the unit vectors in x and y direction. While the mean wave forcing is incorporated through time dependent pressure gradient in x direction in the first term on the right hand side (see equation (2)), the gradient of the dynamic pressure, P′, is given as the second term of the right hand side of equation (11). With the small particle response time and the dilute flow assumptions, the only back effect of sediment on the carrier flow is due to the sediment-induced density stratification in the gravitational direction, i.e., y (the third term on the right hand side of equation (11)). Specifically, this back effect is a function of normalized sediment concentration, C, and the bulk Richardson number. The bulk Richardson number is given as:

equation image

[17] Here, equation image is the normalized volume-averaged sediment concentration which is defined as

equation image

where equation image is the dimensional volume of the computational domain. For a given flow condition and sediment properties, the bulk Richardson number can be considered as the proxy for volume-averaged concentration.

[18] With the approximation shown in equation (9), there is no need to calculate additional sediment phase momentum equations for sediment velocity. As demonstrated by Cantero et al. [2008], the resulting transport of sediment is greatly simplified and can be calculated through the mass conservation:

equation image

The right-hand side of equation (14) contains a diffusive term which serves to approximate the sub-grid scale random particle motion [Cantero et al., 2008]. As pseudo-spectral scheme is used to solve the governing equations, the diffusion term is also required for numerical stability.

[19] The size of the computational box used for the numerical simulations is equation imagex × equation imagey × equation imagez = 60equation image × 60equation image × 30equation image, which approximately corresponds to 10 × 10 × 5 cm in physical space (see Figure 1). The size of the domain is selected to be the same as the one of Spalart and Baldwin [1989] which is the most conservative, i.e., the largest, among similar studies. Ozdemir et al. [2010b] demonstrated that for ReΔ = 1000 the largest eddy size, estimated via two point correlation functions, to be 10equation image and 5equation image in the streamwise and the spanwise directions, respectively. Hence, our choice of computational domain size is sufficient to cover the largest turbulent eddies. This is consistent with our assumption that the flow is fully developed without any bed form considered. The number of grid points in each direction is Nx × Ny × Nz = 192 × 193 × 192. While equally spaced grid points are used in the spanwise and streamwise directions, Gauss-Lobatto collocation points are used in the vertical direction [Canuto et al., 1987]. Through spectral analyses, Ozdemir et al. [2010a, 2010b] showed the grid resolution to be adequate for resolving the entire range of scales, some details regarding the spatial and temporal resolution are also given here. The number of grid points used to resolve the flow within the first Stokes boundary layer thickness, equation image, is 16. In terms of wall units for the turbulent flow cases, the minimum spacing in the vertical direction is Δy+ = 0.18, which corresponds to Δequation image = 0.004equation image, and the maximum spacing is close to the mid-channel and is Δy+ = 22.0 equivalent to Δequation image = 0.5equation image. The distance in wall units can be expressed in terms of both dimensional and non-dimensional units as

equation image

[20] The wall units are obtained by using the maximum frictional velocity, (U)*,max, over the entire wave cycle and is obtained directly from the numerical results of velocity gradient at the no-slip boundary:

equation image

[21] The maximum non-dimensional frictional velocity is found to be 0.052 for cases that shows turbulent characteristics (i.e., regimes 1 and 2, see section 3), and 0.038 for the cases that show characteristics of laminarization (i.e., regimes 3 and 4). The nondimensional timescale of particle settling, equation image0/equation images, approximately ranges between 370 and 3300. The simulations are carried out for several cycles to make sure the equilibrium is achieved as the nondimensional wave period is around 3000. The nondimensional Kolmogorov timescale, (ɛ · ReΔ)−1/2 is found to be 0.791. The nondimensional time step chosen is approximately 0.001, which is sufficient to resolve all the relevant temporal variation.

[22] No slip wall boundary conditions (NSWBC) are imposed for the fluid phase at the top and bottom boundaries of the computational domain. Periodic boundary conditions (PBC) are imposed along the streamwise and spanwise directions. Similarly, for the sediment concentration PBC are implemented in the streamwise and spanwise directions. In the vertical direction the diffusive flux is equated to deposition:

equation image

Hence, the total amount of sediment specified initially is conserved throughout the computation and equation image is constant for each run (see equation (13)). Referring to equation (12), for a given Stokes Reynolds number and sediment specific gravity, the bulk Richardson number Ri is also a constant for each simulation run. In other words, we fix the “sediment availability” in each simulation by choosing the value of Ri. Although this is an idealization of the realistic fine sediment transport, it allows us to avoid addressing the complicated sediment availability issues due to riverine input. Validation of the numerical model and further details of numerical implementation are discussed by Ozdemir et al. [2010a, 2010b].

[23] The set of simulations investigated in this study is summarized in Table 1. All the cases are of the same Stokes Reynolds number of ReΔ = 1000. Cases 1, 4, 7 and 9 are of the same dimensionless settling velocity of 9 × 10−4 but with different bulk Richardson number, Ri. Ri is taken as 1 × 10−4 in most of the remaining simulations. The rest of the simulations are carried out to further investigate the effects of settling velocity. The dimensionless settling velocity ranges from 4.5 × 10−4 to 27 × 10−4, which correspond to mono-dispersed, non-flocculated silt sediment of grain diameter ranging between 17 to 42 μm. Even with this narrow range of particle size, significant variation in the lutocline dynamics can be observed. We shall investigate lutocline dynamics in detail in section 3.

Table 1. List of Simulations Conducted and Their Parameters
CaseReΔRiVsd (μm)Regime
1100009.0 × 10−424Regime 1
210001 × 10−44.5 × 10−417Regime 2
310001 × 10−47.5 × 10−422Regime 2
410001 × 10−49.0 × 10−424Regime 2
510001 × 10−416.0 × 10−433Regime 2 and 3
610001 × 10−427.0 × 10−442Regime 3
710003 × 10−49.0 × 10−424Regime 3
810003 × 10−427.0 × 10−442Regime 4
910006 × 10−49.0 × 10−424Regime 4

3. Results

3.1. Evaluation of Flow Regimes Under Different Settling Velocities

[24] In this section, we discuss the flow regimes through the analysis of iso-concentration contours (Figure 2), and with the use of more quantitative physical characteristics, such as profiles of ensemble-averaged sediment concentration, streamwise velocity and root-mean-square (RMS) of velocity fluctuations (Figure 3).

Figure 2.

(a) The sampled phases (I, II, III) in a wave cycle are defined. The iso-concentration contours of (b) Case 2, (c) Case 3, and (d) Case 6 (i.e., increasing settling velocity under the same sediment availability and Stokes Reynolds number, see Table 1) at three different phases (I), (II) and (III). The sediment concentration value for Case 2, Case 3 and Case 6 is 2.1, 2.5 and 4.0, respectively.

Figure 3.

Ensemble-averaged profiles of sediment concentration, where the top x axis is the concentration scale for Case 6 and the bottom axis is the concentration scale for (a) the remaining cases, (b) streamwise velocity, and (c) RMS of the streamwise velocity fluctuations (c) for three different phases (left to right: ϕ = π/3, π/2, 2π/3). Case 1 is the passive case where sediment has no impact to the carrier flow (regime 1). Case 2 to 4 belong to regime 2 with lutoclines clearly identified. Cases 2 to 4 are of the same sediment availability but with increasing values of settling velocity. Case 6 is of the highest settling velocity where laminarization occurs (regime 3).

[25] The three frames in Figure 2b show the iso-concentration contours at C = 2.1 for Case 2 at wave phases I, II, and III. The graphical representation of wave phase ϕ is defined in Figure 2a. In Figure 2b, small pockets of inclined turbulent structures at I, grow into larger vertically directed clusters at II. These structures then deform into organized long streaks at III. It is noted that the response of the flow turbulence to the wave forcing (mean pressure gradient) plays an important role in the development of turbulence and its persistence. In the studies of both Spalart and Baldwin [1989] and Ozdemir et al. [2010a] an asymmetry in the wall shear stress between the accelerating and the decelerating phases is observed. The adverse pressure gradient is responsible for the rapid growth of turbulence in the decelerating phases [Spalart and Baldwin, 1989]. Therefore, we observe a dense energetic vortex structures at the decelerating phase I, but less energetic long streaks at the accelerating phase III. In the ensemble-averaged concentration profile shown in Figure 3a (green circle for Case 2), the formation of a lutocline, a sharp concentration gradient located around y = 25 is clearly observed. The iso-concentration contours in Figure 2b suggest the flow below the lutocline to remain energetic and turbulent, which can further be confirmed by the ensemble-averaged streamwise RMS velocity fluctuation profiles, Urms shown in Figure 3c. Turbulent velocity fluctuations remain large below the lutocline. Reduction of turbulence compared to the passive case (Case 1) can be observed only above the lutocline. For Case 3 of 67% larger sediment settling velocity, the iso-contours of sediment concentration (see Figure 2c) are qualitatively similar to that of Case 2. From the corresponding ensemble-averaged concentration profile shown in Figure 3a (red square), the formation of a lutocline for Case 3 is also observed, but it is located at a lower elevation (y ∼20) compared to that of Case 2. For Case 4 of even larger settling velocity, iso-concentration of sediment concentration below the lutocline is qualitative similar to that of Cases 2 and 3 (not shown). However, the overall sediment concentration distribution for Case 4 (blue inverted triangle in Figure 3a) is less well-mixed than that of Cases 2 and 3. Essentially, the lutocline feature is more pronounced for the cases of larger settling velocity.

[26] A lutocline, which forms due to sediment-induced density stratification, separates the lower turbulent layer from the upper less turbulent layer. A lutocline is the distinctive characteristic of regime 2 [Ozdemir et al., 2010a]. Because turbulence is only damped above and around the lutocline where turbulence production is already small, very little difference is observed among the ensemble-averaged velocity profiles of Cases 2, 3, and 4 (see Figure 3b). In fact, these profiles are also close to that of the passive case (Case 1). On the other hand, for Case 6 with much increased settling velocity of Vs = 27 × 10−4, the flow is almost laminar throughout the wave cycle except around flow reversal (see Figure 2d). The iso-concentration contours are only perturbed close to flow reversal, i.e., at II, which can be seen in the form of four waves of very small amplitude. These waves grow and form an oblique wave train at III. These waves observed in Figure 2d are formed due to shear instability. The instability characteristics of the waves that are formed at the flow reversal are unclear due to their unsteady nature and rapid decay. The distinction between Kelvin-Helmholtz or Holmboe waves, which are both results of shear instability, requires a careful hydrodynamic stability analysis. A more detailed discussion on the instability is given by Ozdemir et al. [2010a]. The formation of shear instability around flow reversal gives rise to increased RMS velocity fluctuation (Figure 3c), which then decreases soon after III. In regime 3, flow is laminarized due to intense sediment induced density stratification. Although not shown here, in Case 8, where Ri is further increased by factor 3 (Ri = 3 × 10−4; Vs = 27 × 10−4), turbulence is completely suppressed throughout the wave cycle, which falls into regime 4 and the oscillatory boundary layer is completely laminarized due to the presence of large amount of sediment.

[27] Based on the results shown in this section, the existence of four different flow regimes described by Ozdemir et al. [2010a] for different sediment availability (Ri), is also observed to exist when the settling velocity of sediment is varied. Although the damping of turbulence is directly related to sediment concentration through bulk Richardson number, large sediment settling velocity tends to make more sediment accumulate near the bed and hence leads to laminarization. According to the ensemble-averaged concentration profiles for Cases 2 ∼ 4 shown in Figure 3a, the lutocline is lower and the negative concentration gradient is more pronounced as settling velocity is increased. In order to better understand the transition between these flow regimes, we further quantify the lutocline dynamics for different settling velocities in the sections 3.2.13.2.3.

3.2. Dynamics of Lutocline

[28] As discussed in section 2, the signature of the lutocline is the formation of a sharp (negative) gradient in the ensemble averaged concentration profile. This sharp gradient is a transition between two adjacent regions of milder gradient, and is indicated by the existence of an inflection point in the concentration profile (i.e., in the lutocline layer between y2 and y4 shown in Figure 4, we expect the existence of an inflection point d2 (C)/dy2 = 0). Noh and Fernando [1991] also point out this feature in the particle concentration profile suspended by a oscillating grid and state that the inflection point can be considered as the front between the concentrated and dilute regions in the water column.

Figure 4.

A descriptive sketch for the vertical structure of ensemble-averaged sediment concentration profile in regime 2 where a lutocline can be clearly identified. The circles show the upper (y = y4) and the lower (y = y2) borders of the lutocline layer and the circle at y = y3 shows the inflection point. The point (y = y1) is the upper bound of the viscous wall layer.

[29] To facilitate our discussion, a typical sediment concentration profile in regime 2 is illustrated in Figure 4 where the locations of interest y1, y2, y3 and y4 are marked. y1 is the point that defines the extent of the viscous wall region from the bottom wall. In all the simulations that belong to regime 2, we observe y1 = 0.5 ∼ 1.0, which corresponds to y+ = 26 ∼ 52 in wall units.

[30] In Figure 4, y3 is defined as the point of inflection in the ensemble averaged concentration profile (d2 (C)/dy2 = 0). Points y2 and y4 define the lower and upper edges of the lutocline layer. As we discuss in section 4, these points are determined on the basis of the sediment flux budget. The location of y2 is defined as the last point above y1 where the settling flux equals the turbulent suspension flux. Above y2, the sub-grid scale diffusive flux starts to be important. On the other hand, y4 is where the sediment settling flux becomes very close to the sub-grid scale diffusive flux and hence the turbulent suspension flux is negligible.

[31] The existence of the inflection point can be used to explain the lutocline formation and its sustainability. Let us consider the Reynolds-averaged sediment conservation equation:

equation image

[32] Since the Reynolds-averaged mean flow only depends on y (flow is uniform in the streamwise and spanwise directions), the contributions to the advection and diffusion terms from the streamwise and the spanwise directions are zero. The third term on the left hand side is the gradient of ensemble averaged vertical turbulent suspension flux, i.e., 〈C′V′〉. This term can also be considered as turbulent flux due to mixing in the vertical direction. From this point forward, 〈C′V′〉 is denoted as VTSF. At the inflection point, the sub-grid scale diffusion term on the right hand side vanishes. Hence, equation (18) becomes:

equation image

[33] If either the Reynolds-averaged turbulent concentration flux has negligible vertical variation, i.e., ∂ 〈C′V′〉/∂y = 0, or if the flow is not turbulent, i.e., 〈C′V′〉 = 0, equation (19) further becomes an advection equation, or simple wave equation that describes the settling of particles in still water:

equation image

[34] The implication of equation (20) is that, if a vertical concentration profile with an inflection point were to be introduced in a flow which does not support considerable vertical gradient of VTSF, the inflection point will leave the domain with the speed of settling velocity. The concentration profiles in flow regimes 3 and 4 are consistent with the above discussion, since in these two regimes sediments significantly suppress turbulence and the flow becomes laminarized. The concentration profile reduces to laminar solution which uniformly decreases away from the bed and a lutocline does not exist (e.g., see Case 6 in Figure 3a). Moreover, for a very dilute flow in regime 1, where sediment has negligible impact on the carrier flow, the boundary layer is vigorously turbulent and the variation in VTSF becomes very small throughout the boundary layer. Therefore, concentration profile is well-mixed without the features of a lutocline (e.g., see Case 1 in Figure 3a). In essence, the existence of lutocline requires some level of variation in VTSF. In regime 2, the variation in VTSF is the result of turbulence attenuation due to the existence of sediment. Moreover, at least part of the flow needs to remain turbulent. The observations in the literature clearly points out the strong variation in VTSF between the points y2 and y4 [Smith and Kirby, 1989; Wolanski et al., 1988; Noh and Fernando, 1991; Huppert et al., 1995]. Hence, it is clear that the evolution of lutocline, the associated sediment flux, and VTSF are directly related to the transition between the turbulent layer and the less turbulent layer above y4, which we discuss in section 4.

3.2.1. Vertical Structure

[35] The change in the vertical structure, such as the lutocline location and lutocline layer thickness are discussed here with respect to the variation in the settling velocity. From the ensemble averaged sediment concentration profiles (see Figure 3a) it can be seen that the lutocline layer shifts toward the bottom boundary with increasing settling velocity. Although there exists some time dependent variation in the ensemble-averaged concentration within a wave cycle, they are quite small and do not change the main features of the lutocline. For Cases 2, 3, and 4 the concentration within the lutocline layer (between y2 and y4) varies within the range [0.7, 1.7], [0.4, 2.2], and [0.3, 2.6], respectively. The concentration values at the lower end of the lutocline layer (y2) increases with increasing settling velocity. On the other hand, an opposite trend is observed for the top boundary of the lutocline layer (see Figure 5). The lutocline layer for Cases 2, 3, and 4 are located between [y2 = 19, y4 = 28], [16, 27], and [13, 26]. Moreover, the inflection point (y3), which can be considered as the representative location of lutocline, becomes lower for cases of larger settling velocity. In summary, with increasing settling velocity, the location of the lutocline is lower while the thickness of the lutocline layer (y4y2) becomes larger. The variation in concentration together with the variation in lutocline thickness also suggest that the (negative) gradient of concentration is larger for the case of larger settling velocity. Also noted from Figure 3a, the (negative) concentration gradient from the top of the viscous wall region (y1) to the lutocline layer (y2) increases with increasing settling velocity. Qualitatively, damping of carrier flow turbulence due to sediment-induced density stratification is certainly more pronounced for cases with larger settling velocity, which further affects the balance of sediment fluxes, TKE budget and VTSF.

Figure 5.

(a) Averaged concentration values at the upper end of the lutocline layer C (y4), the inflection point C (y3), and the lower end of the lutocline layer C (y2) with respect to different values of nondimensional settling velocity (Vs). (b) The variations of y2, y3, and y4 with respect to different Vs values for Cases 2, 3 and 4.

3.2.2. Sediment Flux Budget and TKE Budget

[36] If the ensemble-averaged sediment conservation equation given in equation (18) is integrated in the vertical direction,

equation image

After imposing the boundary condition given in equation (17), we obtain:

equation image

[37] In addition to sediment flux balance, we also investigate the TKE budget in order to better understand the vertical structure and mixing characteristics in regime 2. The balance equation of turbulence kinetic energy (TKE = k) is presented as follows:

equation image
equation image
equation image
equation image
equation image
equation image

[38] The TKE budget is composed of production, turbulent dissipation, (sediment-induced) buoyancy dissipation, and the transport terms, respectively given on the right hand side of equation (23b)(23f). The buoyancy dissipation term is effective only in the vertical, y, direction. The transport term T(k) is composed of pressure transport, turbulent advection and diffusion term, respectively given in (23f).

[39] The components of sediment flux budget for Cases 2 and 4 are shown in Figures 6a and 7a and the corresponding TKE budget are shown in Figures 6b and 7b. The region close to the bottom boundary (y = 0 ∼ 10) is focused in Figure 6 and the intermediate region that includes the lutocline (y = 10 ∼ 30) is highlighted in Figure 7. Only the results for ϕ = 0 (the phase of maximum positive velocity) during the wave cycle is shown. Although quantitative differences are present between Cases 2 and 4, the characteristics are qualitatively similar. In fact, results for Case 3 are also qualitatively similar and hence are not shown here for brevity. In the sediment flux balance, the vertically integrated time rate of change in concentration is generally smaller than other terms (consistent with the fact the concentration profile does not change much during a wave cycle), and therefore plays a minor role on the overall sediment flux balance. In this study, the settling velocity is constant for each run and therefore the settling flux, −|Vs| 〈C〉|ξ=y, follows the variation of the concentration profile.

Figure 6.

(a) sediment flux budget terms, (b) TKE budget terms, (c) VTSF budget terms and (d) concentration fluctuation variance at phase ϕ = 0, for Cases 2 and 4 in the lower portion of the oscillatory boundary layer (between y = 0 and y = 10). STRC in Figure 6a is the integral time rate of change of sediment concentration.

Figure 7.

(a) sediment flux budget terms, (b) TKE budget terms, (c) VTSF budget terms and (d) concentration fluctuation variance at phase ϕ = 0, for Cases 2 and 4 in the upper portion of the oscillatory boundary layer (between y = 0 and y = 30). STRC in Figure 7a is the integral time rate of change of sediment concentration.

[40] From the bottom wall up to y = y1 = 0.5 ∼ 1.0, turbulent fluctuations are negligible. This can be seen in the TKE budget where the turbulent production is negligible within y < 0.5 ∼ 1.0 and the turbulent dissipation rate is balanced by the transport terms, mainly diffusion. In the sediment flux balance, since turbulence is negligible, the settling flux is counterbalanced by sub-grid scale sediment diffusion. As summarized in Figure 8, this layer (0 < y < y1) is called the viscous wall layer.

Figure 8.

(left) Four flow layers observed in the direction normal to the bed in regime 2 in which a lutocline can be identified. (right) Summary of the dominant terms in the sediment flux, TKE, VTSF and concentration fluctuation variance. In the sediment flux budget, sub-grid scale diffusion is denoted as S.G.S. diffusion. In the quasi-laminar layer, some turbulent statistics cannot be obtained accurately due to limited numbers of realizations because the terms at this layer are essentially not turbulent. These items are indicated with N/A.

[41] VTSF term in the sediment flux budget is more or less balanced by the settling flux above the viscous wall layer (y > 0.5 ∼ 1.0) up to the lower end of the lutocline layer (i.e., y < y2 = 19, 13 for Cases 2 and 4, respectively). In this layer, the integrated time rate of change term occasionally becomes noticeable and somewhat participates in the budget. The sub-grid scale diffusion is negligible in this turbulent region. From the TKE budget, fully turbulent boundary layer features are observed where turbulent production is balanced by turbulent dissipation (see Figure 6b). As summarized in Figure 8, this layer (y1 < y < y2) is called the turbulent layer.

[42] Within the lutocline layer (y2 < y < y4), the sharp decrease of concentration suggests the settling flux must decrease accordingly. In the upper portion of the lutocline layer (y3 < y < y4), the VTSF decreases even more abruptly due to sediment-induced density stratification (see more discussion in section 4), while the sub-grid diffusive flux becomes increasingly more important to balance this defect (see Figure 7a). Evidence on the effect of sediment-induced density stratification can be seen in the TKE budget. In Figure 7b, sediment induced density stratification is about 10 ∼ 20% of the production term at y = 10 and increases in percentage as y further increases (see more detailed discussion in section 3.2.3). Hence, due to the transitional nature of the lutocline layer (y2 < y < y3, see Figure 8), all the terms in the TKE budget and sediment flux budget seem to be important.

[43] Above the lutocline layer (y > y4), the flow is not turbulent and mixing is suppressed across the lutocline layer. This layer is quasi-laminar (see also Figure 8) where settling flux is small (due to low concentration) and is almost completely balanced by sub-grid scale diffusive flux. Because the flow shows quasi-laminar characteristics and is not sufficiently turbulent, higher order turbulent statistics cannot be obtained accurately due to limited number of realizations. However, based on our general understanding on the TKE budget in turbulent boundary layer [e.g., see Pope, 2000], turbulent dissipation is probably balanced by the transport terms above the lutocline layer.

3.2.3. Vertical Turbulent Suspension Flux (VTSF) and Concentration Fluctuation Variance

[44] The damping of turbulence in the lutocline layer is observed both in experimental/observational [Wolanski et al., 1988, 1992; Huppert et al., 1995] and numerical studies [Noh and Fernando, 1991; Ozdemir et al., 2010a]. Based on the momentum equation, i.e., equation (11), it is clear that such damping is due to the coupling term, −Ri · Ce2, which depends on normalized sediment concentration weighted by the bulk Richardson number. As shown in the TKE budget (Figure 7b), sediment-induced density stratification, which is proportional to the VTSF, 〈C′V′〉, becomes important in the lutocline layer. The magnitude of sediment-induced density stratification is only a fraction of the turbulent production term, which is qualitatively consistent with the critical Richardson number concept [Miles, 1961]. VTSF is the most critical term responsible for the upward suspension of sediment. On the other hand, increasing magnitude of VTSF further damps turbulence through the density-stratification term in the TKE budget equation. Hence, VTSF plays a critical and competing role in both the sediment flux budget and TKE budget. Using the simulation results, we would like to further clarify the role of VTSF (〈C′V′〉) in the suspension of sediment and the suppression of mixing using the budget of VTSF and concentration fluctuation variance.

[45] The budget of VTSF is obtained through algebraic manipulation of the momentum equation in the vertical direction and the sediment mass balance equation [Durbin and Reif, 2001]. The resultant equation is shown as follows:

equation image
equation image
equation image
equation image
equation image

[46] The budget is composed of production, P(〈C′V′〉), turbulent dissipation, ɛv (〈C′V′〉), buoyancy destruction, ɛs (〈C′V′〉), and T (〈C′V′〉) representing various transport terms. The transport terms shown in equation (24e) represent turbulent advection, settling transport, pressure transport and sub-grid diffusive transport.

[47] It is important to note that from the destruction term due to density stratification in equation (24d), the variance of sediment concentration fluctuation, 〈C′C′〉, has an impact on the budget of VTSF. Therefore, it is also informative to examine the budget equation of the concentration fluctuation variance. The budget equation of the concentration fluctuation variance is obtained from the sediment mass balance equation as:

equation image
equation image
equation image
equation image

[48] The concentration fluctuation variance budget equation is composed of production (equation (25b)), turbulent destruction (equation (25c)), and various transport terms shown in equation (25d) represented by T (〈C′C′〉). The overall transport terms consist of turbulent transport, settling convection and sub-grid diffusive transport.

[49] The most notable property in the balance equation of concentration fluctuation variance is that the production term (see equation (25b)) is proportional to VTSF (〈C′V′〉). On the other hand, in the balance equation of VTSF, the destruction due to density stratification is a function of concentration fluctuation variance 〈C′C′〉 (see equation (24d)). This interesting property can be used to explain the equilibrium at the lutocline layer. As VTSF increases, the production term in the concentration fluctuation variance also increases. However, increased concentration fluctuation variance can further increase the destruction of VTCF via density stratification and hence inhibit the growth of VTSF.

[50] The various terms in the budget of VTSF and concentration fluctuation variance are shown in Figures 6c and 6d for the region very close to the bottom wall and in Figures 7c and 7d for the intermediate region that includes the lutocline. The equilibrium between VTSF and concentration fluctuation variance is illustrated here using Case 4. In the VTSF budget, the production term and the transport term are more or less in balance in most of the turbulent layer (0.5 < y < 10, see Figure 6c (right)). In the lutocline layer, however, a different picture emerges. The production is more or less balanced by destruction due to density stratification while the transport term becomes less important (see Figure 7c (right), 13 < y < 26). If we take a look at the 〈C′C′〉 budget, the destruction and the production terms are the dominant terms in both the turbulent layer and the lutocline layer (Figures 6d and 7d). More importantly, the locations of the peak value of the production term in the 〈C′C′〉 budget and the buoyancy destruction term in the VTSF budget are quite close to each other in the lutocline layer (i.e., at y = 17 and y = 19, respectively). This illustrates how the equilibrium in sediment flux and TKE budget is achieved in regime 2 in the lutocline layer.

[51] At the level of the present governing equations, VTSF directly modifies sediment concentration via equation (14). The resultant sediment concentration affects the flow field through the momentum equation (equation (11)), i.e., −Ri · Ce2, and eventually changes the turbulence level in the lutocline layer and the VTSF itself. Hence, a simple and adequate two-way coupling between the sediment and the carrier turbulent flow, appropriate for fine sediment, is achieved through the −Ri · Ce2 term in the momentum equation. Moreover, although not explicitly shown here, the role of settling velocity is to change the gradient of concentration via the sediment conservation equation, i.e., equation (14), which further affects the spatial distribution of density stratification term in the momentum equation. As shown previously, the change in settling velocity quantitatively changes the location and concentration gradient in the lutocline layer and ultimately causes transition to either more laminar (regime 3) or well-mixed (regime 1) regimes.

4. Discussions

4.1. Flow Layers in Regime 2

[52] The simulations that are categorized as regime 2 (i.e., Cases 2, 3, 4) exhibit clear features of a lutocline. From an investigation of the ensemble averaged sediment concentration profile, sediment flux balance, TKE budget, budget of VTSF, and the budget of concentration fluctuation variance (Figures 6 and 7), it is clear that there exist four distinct layers that characterize the vertical sediment motion in regime 2 (see Figure 8). The layer closest to the bed (y = 0 ∼ y1) is the viscous wall layer in which the dominant mechanisms in the sediment flux are (sub-grid scale) diffusion and settling. Because it is so close to the bed, the flow in the viscous wall layer is not turbulent and both VTSF and concentration fluctuation variance are negligible. As expected in the TKE budget, the dissipation is balanced by the transport terms (mainly diffusion [Pope, 2000]). Further upwards, boundary layer turbulence starts to dominate. This is represented in the sediment flux balance in the form of drastically increased VTSF, which counterbalances settling. Hence, the layer between y1 and y2 is called the turbulent layer. The practical importance of this layer is that due to high turbulence and hence high carrying capacity, a significant amount of sediment is suspended in this layer. It is also likely that considerable density anomaly can create gravity flow [Wright and Friedrichs, 2006]. The fully turbulent nature in this layer can also be seen in the TKE budget that the turbulent production is balanced by turbulent dissipation. The drastically increased VTSF, 〈C′V′〉, can be understood from the VTSF budget in this layer where production term is balanced by the transport term. Notice that in the VTSF budget, both the dissipation due to viscous effect and destruction due sediment-induced buoyancy effect, which is proportional to concentration fluctuation variance, remain small in the majority of the turbulent layer (y1 < y < ≈ 10). We also point out here that although the production of 〈C′C′〉 is proportional to 〈C′V′〉, it remains small due to very low vertical concentration gradient (see equation (25b)).

[53] In the lutocline layer (between y2 and y4), where the formation of a sharp negative concentration gradient is located, a two-way coupled state between sediment and flow turbulence exists. In the sediment flux budget, we observe decrease in VTSF when moving away from the bed, which is taken over by increasing (sub-grid scale) diffusion. Among the TKE budget terms, turbulent dissipation and production remain the two largest terms while increasing role of sediment-induced buoyancy and transport terms can also be observed. Essentially, the lutocline layer is the transitional layer where flow and sediment transport characteristics transform as we move from the lower turbulent layer to the upper laminar layer. Hence, all the terms in the TKE budget are of importance. Suppression of VTSF and mixing are more clearly seen through the budget of VTSF. The production term in the VTSF budget decreases in the lutocline layer when moving upward and sediment-induced buoyancy effect becomes the most important. The reason for this large sediment-induced buoyancy effect is due to large concentration fluctuation variance, which has large production in the lutocline layer. Interestingly, this large production is due to large VTSF and concentration gradient in the lutocline layer. Hence, there is a close coupling and equilibrium between the VTSF and concentration fluctuation variance which clearly contributes to the formation of lutocline. Above the lutocline layer (y > y4), a quasi-laminar flow layer is observed. VTSF becomes negligible while diffusive flux counterbalances with the settling flux. The laminar characteristic above the lutocline is discussed in literature for tidal boundary layer [e.g., Wolanski et al., 1988]. Similar features are observed here for an oscillatory boundary layer.

4.2. Onset of Laminarization

[54] A closer look at the simulation results of Case 5 (with a settling velocity of Vs = 16 × 10−4) shows characteristics of both regimes 2 and 3 (Figure 9). From the time series of the plane-averaged near bed sediment concentration at y = 4 × 10−3, i.e., the first grid point above the bed (see Figure 9a), concentration continues to increase even after peak free-stream velocity and reaches its maximum around flow reversal (ϕ = 2π ∼ 2.5π). Near bed sediment concentration remains large until about ϕ = 2.8π. The increase of near bed concentration even after peak flow velocity is one of the main characteristics of laminarization. This is further verified by the iso-concentration contours shown in Figure 9b and 9d at flow peak (ϕ = 2π) and flow reversal (ϕ = 2.5π), respectively. At instant in Figure 9b, we observe more or less a quiescent flow field, which then turns into a sequence of four waves at the time of flow reversal (see Figure 9d), a typical signature of regime 3. Comparing the plane-averaged sediment concentration profiles at peak flow Figure 9b and flow reversal (see Figure 9f), we observe the disappearance of lutocline (y ≈ 10) and a significant increase of near bed sediment concentration (y = 0 ∼ 3) due to settling as flow proceeds from peak flow to flow reversal.

Figure 9.

(a) Time series of the plane-averaged near bed concentration of Case 5 (solid line) along with time series of free stream velocity shown as dashed line. (b and d) The iso-concentration contours (at C = 6.0) at ϕ = 2π and ϕ = 5π/2, respectively, during the phase with the features of laminarization (regime 3). (c and e) Those during the next wave at ϕ = 4π and ϕ = 9π/2 with turbulent features (regime 2). (f) Plane-averaged concentration profiles at ϕ = 2π, 5π/2 and 3π during the laminar wave phase and (g) those at ϕ = 4π, 9π/2, and 5π.

[55] The weak lutocline feature observed during Figure 9b is reminiscent of turbulent transport (regime 2) in the previous wave phase (ϕ = 0 ∼ 2π). This can be further illustrated in detail for the wave phase between (ϕ = 4π and ϕ = 6π). According to the time series of near bed sediment concentration (see Figure 9a), the increase of concentration during ϕ = 3.5 ∼ 4π is weaker than that of the previous wave. The near bed concentration starts to decrease right after the peak flow due to strong turbulent suspension. This can be verified by the iso-concentration contours shown in Figure 9c and 9e. While a small strip of the domain seem to be quiescent (z = 3 ∼ 10) initially, the growth of boundary layer turbulence during deceleration is clearly observed in other parts of the domain (see Figure 9c). Later during flow reversal (see Figure 9e), the intense turbulence takes over the entire domain. Comparing the plane-averaged concentration profiles between the peak flow (Figure 9c) and the flow reversal (Figure 9e) (see Figure 9f), we observe significant decrease of near bed concentration and the initial appearance of lutocline at y ≈ 7 at Figure 9e. This observation suggests strong turbulent suspension transports sediment upward, which is consistent with the turbulent iso-concentration contours shown in Figures 9c and 9e. Moreover, a comparison of plane-averaged concentration profiles between Figures 9e and 9g further suggests the near bed sediment concentration start to increase in Figure 9g, indicating the onset of laminarization for the next wave cycle.

[56] Observations made in Case 5 let us conclude that the transition between regimes 2 and 3 due to increasing settling velocity is not abrupt. The mixed characteristics of both regimes 2 and 3 in Case 5 lead us to the following conjecture. In Figures 3 and 5 it was observed that with increasing settling velocity in regime 2 the lutocline layer moves down toward the bottom boundary. At sufficient settling velocity it appears that the lutocline layer begins to interfere with turbulence production (turbulent layer) and results in laminarization of the flow. As the flow laminarizes the lutocline cannot be supported anymore and the concentration profile approaches the laminar profile. The resulting density gradient is however not strong enough to suppress flow instability dominant during flow reversal and as a result disturbance grows and intermittent turbulence is generated, which in turn supports resuspension of particles toward a lutocline profile. In case 5, the condition is such that the flow switches from being laminarized in one wave cycle to being highly unstable during the next, thus exhibiting features of both regimes 2 and 3.

4.3. Saturation/Laminarization Criterion

[57] In regime 1, flow is turbulent throughout the boundary layer and sediments are almost entirely carried by turbulence. Majority of sediment in regime 2 is carried below the lutocline where the flow remains turbulent. The duration of turbulence generation around flow reversal in regime 3 is short and is of limited effect on transport. Therefore, the amount of sediment conveyed through turbulence can be neglected in regimes 3 and 4. Hence, regime 2 can be considered as a transitional regime where the carrying capacity by turbulence is of significance due to increase in sediment availability and settling velocity. This observation is in coherence with the concept of saturation concentration for sediment transport [Winterwerp, 2001, 2006; Bagnold, 1956]. Using a Reynolds-averaged numerical model with a k-equation image closure that incorporates sediment-induced density stratification, Winterwerp [2001] investigated fluid mud transport in the tidal boundary layer where he predicted the formation of lutocline and subsequent collapse of turbulence and rapid sedimentation during slack tide. The sudden collapse of turbulence when tidal flow is less energetic is explained by a saturation condition, which depends on sediment concentration, flow velocity, equation image and settling velocity equation images [Winterwerp, 2001]:

equation image

where equation image is the flow depth, Ks is an empirical parameter and equation images is the minimum concentration that causes this collapse, defined as saturation concentration. Drawing an analogy to the work of Winterwerp [2001], the transition observed here between regime 2 and regime 3 for fine sediment transport in oscillatory (wave) boundary layer can also be expected to satisfy a saturation criterion. Our main objective here is to evaluate such criterion from several different perspectives.

[58] A measure of flow stability and laminarization can be examined through flux Richardson number, RiF. It is the negative ratio of the buoyancy destruction term, RiCUi〉, to the turbulence production term, −〈UiUjequation image, in TKE budget equation. The flux Richardson number profiles at the peak velocity (ϕ = 0) for Cases 2, 3 and 4 are shown in Figures 10a, 10b and 10c, which cover the upper section of the turbulent layer and the lower section of the lutocline layer. The dashed lines in Figure 10a10c represent the location of y2 for each case. When the flow is not sufficiently turbulent, the flux Richardson number increases drastically, such as that above y2. For Case 2 of the lowest settling velocity, the flux Richardson number in the turbulent layer is around 0.08 (Figure 10a). For the cases with larger settling velocity the minimum value of RiF increases (see Figures 10b and 10c). In a steady flow, the critical Richardson number for flow instability, above which flow is stable, is theoretically shown to be 0.25 [Miles, 1961; Howard, 1961]. If we take this critical value to be relevant in the present oscillatory case as well, we can roughly estimate the unstable region to be the layer where local RiF is below 0.25. If we choose the minimum value of RiF in the turbulent layer as the representative value for each case and plot this minimum value against non-dimensional settling velocity (see Figure 10d), we observed roughly a linear trend. By extrapolating to RiF = 0.25, the corresponding non-dimensional settling velocity for saturation condition is about Vs = 0.0012. As discussed in section 4.2, in Case 5 where Vs = 0.0016, we observe alternating transition between regime 2 and 3 at different wave cycles, which can be considered as the onset of laminarization. The extrapolation line for Case 5 corresponds to RiF = 0.33. Therefore, based on the simulation results, a conservative estimate suggests a critical flux Richardson number to exist between [Vs = 0.0009, RiF = 0.18] and [Vs = 0.0016, RiF = 0.33]. Results here are much in coherence with the analysis of Turner [1973] where a critical RiF, before the collapse of turbulence, exists between 0.05 and 0.30, based on laboratory and field experiments. Field observations of interfacial instability under a river plume also suggest a critical Richardson number of about 0.33 [Geyer and Smith, 1987].

Figure 10.

Vertical profiles of flux Richardson number, RiF, in the upper portion of the turbulent layer and the lower portion of the lutocline layer for (a) Case 2 (Vs = 4.5 × 10−4); (b) Case 3 (Vs = 7.5 × 10−4) and (c) Case 4 (Vs = 9.0 × 10−4). The location of y2 in each case is shown as the dashed line. (d) Variation in the minimum of RiF with respect to settling velocity, Vs. Two dashed lines in Figure 10d indicates the corresponding nondimensional settling velocity associated with Ri = 0.25 and 0.33 based on the linear extrapolation.

[59] Second, the amount of sediment carried by turbulence in regime 2 can be evaluated by integrating sediment concentration in the turbulent layer as

equation image

[60] The turbulent layer is chosen to provide an estimate of the sediment carrying capacity resulting from a balance between VTSF and sediment settling in the sediment flux balance. Variation in B due to different settling velocity (Cases 2–5) is shown in Figure 11. The results are in reasonable agreement with the concept of saturation concentration. For the three lower settling velocities (Cases 2 to 4), we observe a more or less constant value of B. However, among these three cases, the value of B calculated for Case 4 with the largest settling velocity (Vs = 9.0 × 10−4) is noticeably lower, suggesting a downward trend of B as the settling velocity increases. As discussed in section 4.2, Case 5 exhibits a transitional behavior between regimes 2 and 3. Namely, one observes a turbulent flow feature under a given wave cycle and then followed by a non-turbulent (laminar) wave cycle. Here, Figure 11 shows B for Case 5 averaged only during the period when the flow exhibits turbulent lutocline features of regime 2 (shown as filled circle). The resulting B value is about 25 to 30% smaller to that of Cases 2–4, and the lower value is due to the fact that the flow is turbulent only part of the wave cycle. However, when B is evaluated by averaging over the entire duration of simulation (including both the turbulent and non-turbulent periods), the resulting B is significantly lower by a factor of 3 plotted as open symbol. Although the number of cases performed here is limited, by linearly extrapolating the points associated with Cases 3 and 5 (see dashed line) to B = 0, the critical nondimensional settling velocity for laminarization (i.e., saturation) can be approximately estimated to be around 0.002.

Figure 11.

Variation in the carrying capacity, B, with respect to nondimensional settling velocity (for Cases 2, 3, 4 and 5). For case 5, B is evaluated through time averaging over only the time period when lutocline exists (shown with solid circle) and during the entire period of simulation (shown with open circle). The solid line is the linear fit of Cases 3, 4 and 5 when Case 5 is averaged during the times when lutocline is observed. The dashed line is the linear fit of B when Case 5 is averaged over the whole simulation time. The interception of the dashed line with B = 0 (x axis) can be used to qualitatively estimate the critical nondimensional settling velocity for laminarization.

4.4. Flow Regimes in the Parametric Space

[61] Our capability to predict the flow regimes in a 3D parametric space of Reynolds number, Richardson number and non-dimensional settling velocity is extremely important for many oceanic applications. As described briefly in section 1, the transition from regime 1 to regime 2, i.e., from well-mixed transport to lutocline development, implies the formation of suspended fine particles in the wave boundary layer and hence major offshore sediment transport may occur, e.g., through wave-supported gravity flow [e.g., Wright and Friedrichs, 2006]. In addition, the transition from regime 2 to regime 3 or regime 4 implies laminarization of a wave-induced suspended fine particle transport, which has several implications. The most important of all is the calculation of wave-induced fine particle transport may be much simpler because flow is not turbulent and hence laminar solutions may be applicable. Also, recent field observations suggest this to be the regime where significant attenuation of surface wave energy is often observed [e.g., Traykovski, 2010]. Hence future research emphasis should perhaps focus on the non-Newtonian behavior of the laminar mud layer due to high sediment concentration and how such mud response may attenuate surface wave energy transfer (among different wave frequency) and energy destruction [e.g., Dalrymple and Liu, 1978; Mei and Liu, 1987].

[62] According to the present governing equations, it is clear that the flow regimes must depend on the Stokes Reynolds number (ReΔ) bulk Richardson number (Ri) and nondimensionalized settling velocity (Vs). We propose to write the general form of a flow regime predictor as

equation image

where f0 is a function of bulk Richardson number, Ri, and the settling velocity of the single particle in still water, Vs. K0 is a function denoted by f1 which depends on ReΔ [Hino et al., 1983]. Since in the present study we investigate 9 cases (see Table 1) holding Reynolds number constant at ReΔ = 1000, we consider f1 to be constant and equal to K0. It is interesting to point out that the saturation criterion for fluid mud in tidal boundary layer shown in equation (26) [see also Winterwerp, 2001] is consistent with the general form shown in equation (28) (replacing the length scale in the bulk Richardson number with the flow depth for tidal boundary layer). For tidal boundary layer, the flow is fully turbulent and the dependence on Reynolds number may be negligible. Hence, the right-hand-side of equation (26) can be taken to be a constant. Making an analogy to tidal boundary layer, the saturation condition in the present wave boundary layer condition of constant Stokes Reynolds number is assumed to satisfy

equation image

[63] The above simplified form of f0 can be justified for the present study by the following argument. The only coupling term between the sediment phase and the carrier fluid flow that causes the different flow regimes is through the third term on the right-hand-side of the momentum equation (equation (10)), i.e., −Ri · C. Since the flow remains laminar in regimes 3 and 4, sediment concentration profile can be taken to be close to the laminar solution. In the present configuration, the laminar solution of the concentration profile attains a value of 60 VsReΔSC at the bottom wall [see Ozdemir et al., 2010a]. Hence, at the bottom wall, the coupling term becomes −60 VsReΔSC. For constant ReΔ and SC, a criterion for saturation can be quantitatively represented by equation (29). Figure 12 presents all the 9 cases considered in this study in the parametric space of Ri versus Vs. The four flow regimes are also shown with the borders between each regime represented by equation (29) with the corresponding K0 approximately determined by the flow regimes of the 9 simulations. Since simulation results of Case 5 shows transitional features, the border between regimes 2 and 3 (K0 = 1.6 × 10−7) is drawn to pass through case 5. The border between regimes 3 and 4 is approximately depicted to be given by K0 = 5.5 × 10−7. However, as mentioned in section 4.2, the transition between regimes 2 and 3 is not expected to be abrupt. Rather, there exists a transitional area in the parametric space that might lead to intermittent occurrence of lutocline.

Figure 12.

The map of the regimes observed in 2D parametric space defined by Ri and Vs at ReΔ = 1000. The conducted 3D simulations (see Table 1) are shown as solid dots. Additional 1DV Reynolds-averaged model runs [Hsu et al., 2009] are shown as open triangles. The dashed curves signify qualitatively the borders between the regimes. Moreover, the actual transitions between regimes are not abrupt, such as that discussed in Figure 9.

[64] The limited number of runs, which are carried out here, cannot allow us to firmly pinpoint the borderline between regime 1 and regime 2. Using the one-dimensional Reynolds-averaged numerical model of Hsu et al. [2009], which is much more computationally efficient than the present 3D numerical simulation, several runs were carried out at ReΔ = 1000. Based on the Reynolds-averaged model results (triangle in Figure 12), we observe that the transition between well-mixed regime 1 and lutocline formation in regime 2 is rather gradual when settling velocity and/or sediment availability are increased. Following the nondimensional parameters used here, clear lutocline can be seen at around Vs = 2 × 10−4 for fixed Ri = 1 × 10−4. Moreover, for fixed Vs = 9 × 10−4, clear lutocline can be seen for Ri ∼ 1 × 10−5 (near bed concentration below O(1) kg/m3).

5. Relevance to Field and Laboratory Observations

[65] As discussed in section 3.1, the simulation results reveal the existence of four different flow regimes for a range of settling velocity. These flow regimes are similar to earlier simulations reported by Ozdemir et al. [2010a] for a range of sediment availability. Although sediment availability is kept constant, the effect of settling velocity is realized through changes in the concentration profile, which modifies the sediment-induced density stratification. The increase in settling velocity lowers the lutocline location and gives rise to increase in concentration near the bed and eventually hinders turbulent production in the turbulent layer. The predicted shape of concentration profiles in regime 2 and its transition to a laminar profile in regimes III and IV are similar to those observed by Lamb et al. [2004] in a laboratory U-tube.

[66] One important simplification in the present simulations is the assumption of a monochromatic oscillatory forcing, which is highly simplified when compared to wave forcing observed in the field. In fact, in most field observations mean currents are likely to co-exist with waves. However, field observations reported by Traykovski et al. [2000, 2007] and Traykovski [2010] suggest high concentration mud suspension or fluid mud (similar to regime 2 ∼ 4 in the present study) often exist under relatively weak current conditions. Strong currents are likely to make the sediment well-mixed in the water column. It is also worth mentioning that the field observed sediment concentration profiles reported by Traykovski et al. [2000, 2007] that are called “fluid mud” events often do not show a pronounced shoulder shape (i.e., no clear inflection point, see Figure 4 between y2 to y4). Hence, it is likely that laminarization of wave boundary layer due to the presence of fluid mud is a ubiquitous feature. More field evidences and data analyses on laminarization are certainly necessary.

[67] The existence of the flow regimes described both with variable settling velocity, Vs, and the sediment availability, Ri, gave rise to a preliminary demarcation of the flow regimes in 2D parametric space defined by Vs and Ri for constant ReΔ. It must be cautioned that such a regime map is expected to be highly dependent on ReΔ. Moreover, the numerical model currently does not include critical mechanisms of cohesive sediment such as flocculation, non-Newtonian behavior, and hindered settling which are inherently existent in fluid mud transport of high sediment concentration. Nevertheless, we believe the onset of laminarization investigated here, which is mainly controlled by turbulence-sediment interaction, can still provide a useful guideline for future study of more realistic fluid mud transport.

[68] Under field conditions, sediments are of polydisperse nature. To the best of our knowledge, the turbulence-resolving numerical studies that aim to address polydispersed sediment transport in the wave boundary layer are scarce. As long as the sediment contains only fine particles such that their Stokes number is much smaller than unity, we believe the results obtained here may be qualitatively applicable to polydispersed conditions as well (e.g., using an averaged settling velocity). However, when the sediment contains fine to medium sand (grain size greater than 100 μm), where the Stokes number may be greater than one, a more complete model that accurately accounts for the particle phase velocity is required. Extending the present numerical model to coarser particles and a polydispersed formulation is certainly a critical future work.

6. Concluding Remarks

[69] Turbulence resolving numerical simulations are conducted for different settling velocities to study the evolution of lutocline subject to the same sediment availability (bulk Richardson number) and oscillatory flow condition (Stokes Reynolds number). As a result of these simulations, we have identified the four flow regimes previously shown to exist for different sediment availability [Ozdemir et al., 2010a], to also exist under different settling velocities. Here, special attention is given to the dynamics of lutocline observed in regime 2 as it gives critical insights into the transition between the flow regimes. Based on the ensemble-averaged sediment concentration and velocity profiles and budgets of sediment flux and TKE, we categorize the vertical flow structure in regime 2 as (1) the viscous wall layer where flow is not turbulent due to its proximity to the bed; (2) the turbulent layer where VTSF is dominant and major transport of sediment takes place; (3) the lutocline layer where sediment-induced density stratification begins to be effective and the dominance of turbulence starts to drastically decay which eventually gives rise to (4) the quasi-laminar flow layer where the flow is essentially laminar. Simulation results in regime 2 suggest that when increasing the settling velocity, the location of the lutocline layer moves closer to the bed, which essentially suppresses the turbulent layer, and flow eventually laminarizes (regimes 3 or 4) when settling velocity increases above a threshold.

[70] Based on the TKE budget and the flux Richardson number, it is observed that the sediment induced density stratification is small in the turbulent layer but becomes effective in the lutocline layer. In regime 2, the flux Richardson number is significantly smaller than 0.25 near the bed, however, above the upper section of the turbulent layer, flux Richardson number increases sharply to exceed 0.25 in the lutocline layer. The role of the lutocline layer that can effectively suppress mixing and separate the lower turbulent layer from the upper quasi-laminar layer is further illustrated with the budgets of VTSF and concentration fluctuation variance. Balance equations of these two second-order turbulence statistics are derived and it is evident that in the balance equation of VTSF, the damping term due to sediment-induced buoyancy, which peaks at the lutocline layer, is proportional to concentration fluctuation variance. On the other hand, the production of concentration fluctuation variance, which also peaks at the lutocline layer, is proportional to VTSF and vertical gradient of sediment concentration. Hence, the interplay between VTSF and concentration fluctuation variance must reach certain equilibrium at the observed lutocline layer.

[71] The concept of saturation for fine sediment is further investigated here for wave boundary layer. Based on the vertically integrated carrying capacity and minimum flux Richardson number in the turbulent layer, a critical settling velocity for saturation (transition from regime 2 to regime 3) can be estimated through extrapolation. However, numerical simulation specifically carried out to study the onset of laminarization (Case 5) show hybrid flow features of regimes 2 and 3 which alternate between different wave cycles, suggesting that the transition to laminarization is not abrupt. For practical applications, we attempt to develop a predictor for the observed flow regimes in the parametric space of bulk Richardson number, non-dimensional settling velocity and Stokes Reynolds number. For the present simulations under the constant Stokes Reynolds number of 1000, we propose a functional relationship based on rational arguments of theoretical laminar concentration profile and the coupling term in the momentum equations. Using the nine simulations carried out so far, we reveal a two-dimensional flow regime map that can be useful toward predicting the state of muddy seabed in moderately energetic continental shelves.

[72] It is worth mentioning that mud often exists in the coastal environments that are not very energetic. The simulations conducted heretofore are idealized condition for the most energetic muddy shelve where the Stokes number is ReΔ = 1000. Because the Reynolds number relevant to muddy environment is usually low and falls within the perturbed laminar or intermittently turbulent regimes, the resulting flow and sediment transport must strongly depend on the Reynolds number. Hence, more simulations at lower Stokes Reynolds number range are necessary.

Acknowledgments

[73] This study is supported by the U.S. Office of Naval Research (N00014-11-1-0270) and National Science Foundation (OCE-0913283; OCE-0926974) to the University of Delaware (OISE-0968313) to the University of Florida. S.B. thanks support from NSF through grant OISE0968313. This work is partially supported by the National Center for Supercomputing Applications under OCE70005N and OCE80005P utilizing the NCSA Cobalt and PSC Pople. Also, the authors acknowledge the University of Florida High-Performance Computing Center for providing computational resources and support that have contributed to the research results reported in this paper.