Journal of Geophysical Research: Oceans

A numerical investigation of convective sedimentation

Authors


Abstract

[1] Understanding the fate of riverine sediment in the coastal environment is critical to the health of the coastal ecosystem and the changing morphology. One of the least understood mechanisms of initial deposition is the convective sedimentation of hypopycnal plumes. This study aims at investigating convective sedimentation by means of a numerical model for fine sediment transport solving the non-hydrostatic Reynolds-averaged Navier-Stokes equations for stratified turbulent flow. Model validation is sought by comparison to laboratory results for turbidity and saline currents over a changing slope. The model is shown to be capable of predicting both the upstream supercritical and the downstream subcritical flows. The numerical model is then utilized to study convective sedimentation and its depositional and mixing characteristics. By analyzing model results of more than 40 runs for different inlet sediment concentration (density ratio γ), settling velocity (particle Reynolds number Rep), and inlet velocity/height (inlet Reynolds number Re), four distinct flow regimes are revealed. For large γ, we observe divergent plumes with significant deposits near the inlet. For intermediate γ and large Rep, intense convective fingers are predicted which are only marginally affected by ambient shear flow. Further reducing the density ratio γ or Rep gives weak convective fingers that are significantly affected by the ambient shear flow. Eventually, no convective fingers are observed during the computation for very small γ or Rep. Sediment deposits in the divergent plume and intense convective finger regimes are relatively insensitive to Re. Deposit increases with Re in the weak convective finger regime.

1. Introduction

[2] Studying the fate of terrestrial sediment in the coastal ocean is critical to our better understanding of the health of the coastal ecosystem [e.g., Fabricius and Wolanski, 2000; Sherwood et al., 2002], coastal morphodynamics [e.g., Wright, 1977; Friedrichs and Wright, 2004] and carbon sequestration [e.g., Kao and Liu, 1996; Goldsmith et al., 2008]. The initial deposition of river-borne sediment and the subsequent sediment transport pathway can be qualitatively investigated according to sediment concentration in the river plume [Wright and Nittrouer, 1995; Geyer et al., 2004]. For most rivers in the world, including some small mountainous rivers, the river-borne sediment concentration rarely exceeds critical concentration for hyperpycnal flow (about 40 g/l) [e.g., Mulder and Syvitski, 1995], making the sedimentation process of a hypopycnal plume the main interest. To predict the location of initial sediment deposition, it is important to estimate the effective settling velocity of sediment in the salt-stratified sediment-laden plume. Due to the fine-grained nature of the river-borne sediments, estimating the settling velocity as primary particle using Stokes' law often gives a value of no more than O(0.1) mm/s. In addition, it is well-known that flocculation processes can significantly enhance sediment settling velocity by one or two orders of magnitude (∼1 mm/s) [e.g., Hill et al., 2000].

[3] Field observations at the mouth of Santa Clara River (CA) by Warrick et al. [2008] capture high concentration (>10 g/L) gravity-driven fluid mud transport near the seabed within 4∼6 h after river discharge events. The fluid mud events are observed through bottom tripod located at about only 1 km offshore of the river mouth. Hence, some very rapid sediment settling mechanisms, which give an effective sediment settling velocity of 1∼2 cm/s, must occur at the river mouth. These limited but valuable field evidences imply that when sediment concentration in the river plume exceeds ∼O(10) g/L during episodic river flooding, effective settling velocity can exceed a few cm/s, which cannot be explained using conventional Stokes settling law based on primary particles or flocs [see also Warrick et al., 2004]. In the conceptual model by Warrick et al. [2008], such rapid settling of river-borne sediment may cause a divergent disposal system, that is, a surface sediment-laden plume and a high concentration bottom gravity flow, which further affects the resulting sediment deposition patterns.

[4] Convective instability across the density interface [e.g., Green, 1987; Parsons et al., 2001] may be the main mechanism causing such rapid sedimentation. Let us first consider fine sediment initially well-mixed and confined in a layer of lighter fluid (e.g., fresh water) that is on top of a sediment-free, denser fluid layer (e.g., seawater). Many prior laboratory experiments indicate instabilities can occur near the density interface, forming finger-like convection that drives rapid sedimentation [e.g., Green, 1987; Hoyal et al., 1999a; Parsons et al., 2001; McCool and Parsons, 2004]. When there is no ambient stratification (i.e., ambient flow is of homogeneous density), the occurrence of convective fingers of fine sediment is a well-known phenomenon, which essentially a form of particle-laden Rayleigh-Taylor instability [e.g., Bradley, 1965; Bush et al., 2003]. In stably stratified still water, convective fingers can be triggered by a double-diffusive mechanism [e.g., Green, 1987; Chen, 1997; Hoyal et al., 1999b; Parsons and García 2000] or a settling-driven mechanism [e.g., Hoyal et al., 1999a]. Essentially, both mechanisms can effectively displace sediments from the less dense upper layer to the denser lower layer and then allow Rayleigh-Taylor instability to trigger rapid settling. According to estimations of settling flux and double diffusive flux suggested by Green [1987], Hoyal et al. [1999a] demonstrate the dominant mechanism causing convective fingers in a typical salt-stratified river plume is due to settling-driven convection. Additionally, in a stratified shear flow, which is more relevant to the flow condition of a river plume, convective sedimentation can be enhanced by (or dominated by) the shear flow [Maxworthy, 1999; McCool and Parsons, 2004]. Essentially, interfacial mixing in a stratified shear flow, possibly through interfacial instabilities [e.g., Geyer and Smith, 1987], can also effectively carry sediment from the lighter upper layer to the denser lower layer. Maxworthy [1999] first demonstrates interfacial mixing induced convective fingers through laboratory experiments. The size of such convective finger is generally on the order of several centimeters, which is significantly larger than that developed in still water. McCool and Parsons [2004] carry out a series of laboratory experiments with different inlet flow velocities and confirm that convective sedimentation and convective finger size are proportional to the inlet velocity, which controls the level of interfacial mixing in the stratified shear flow. However, there is no direct field data demonstrating the occurrence of convective sedimentation or instability.

[5] Observations made at the mouth of Santa Clara River (CA) by Warrick et al. [2008] are based on bottom tripod measurements, not direct measurement of sediment fingers in the water column. Nevertheless, based on the magnitude and timing of measured near bed salinity and turbidity, they conclude that the observed rapid settling may be explained by convective sedimentation. While it is extremely difficult to carry out shipboard measurement immediately after a river flooding event to capture the occurrence of convective sedimentation, laboratory experiment may also suffer from scale effects. Moreover, because the total sediment deposition in a laboratory-scale convective sedimentation experiment is in fact a very small amount, there has been no attempt to collect and quantify the total deposit for different runs.

[6] Although it is not clear what is the typical size of the convective fingers for the field scale, limited laboratory evidence suggests that typical size of the convective fingers is likely not greater than O(10) cm [Maxworthy, 1999]. Hence, it is quite obvious that convective fingers cannot be resolved by typical coastal modeling system [e.g., Warner et al., 2008]. Indeed, existing coastal modeling systems often adopt large grid size and hydrostatic pressure assumption in order to resolve large-scale hydrodynamic and morphological processes. Flow instability is certainly a very small-scale non-hydrostatic process and a more detailed numerical study is necessary. Here, a two-dimensional vertical (2DV) non-hydrostatic Reynolds-Averaged Navier-Stokes (RANS) numerical model for salt-stratified positively buoyant sediment-laden flow is developed and utilized to study convective sedimentation.

[7] This paper is organized as follow. The model formulation and its numerical implementation are summarized in section 2.1, followed by a model validation with laboratory experiment on turbidity currents and saline currents reported by García [1993] in section 2.2. An idealized numerical flume for the problem of convective sedimentation in a river plume is described in section 3.1 along with a discussion of relevant nondimensional parameters. Section 3.2 is devoted to analyses and discussions of the numerical results for the occurrence of convective instability and the resulting deposits. This study is concluded in section 4.

2. Numerical Model

[8] The mathematical formulation and numerical implementation of a non-hydrostatic 2DV Reynolds-averaged Navier-Stokes model for fine sediment transport in a salt-stratified environment is presented in this section. A more detailed discussion of its governing equations, closures, and boundary conditions, as well as its application to model wave-mud interaction can be found in work by Torres-Freyermuth and Hsu [2010].

2.1. Model Formulation

2.1.1. Governing Equations

[9] A measure of the sediment particle inertia effect on the carrier flow can be estimated by the particle response time

equation image

where μ is the dynamic viscosity of the interstitial fluid, ρs is the particle density and f (ϕ) is a function representing the hindered settling which is further dependent on the particle volume concentration ϕ. In this study, we consider typical river-borne sediment of grain diameter D no more than 80 μm and the associated particle response time is no more than O(10−3) sec, suggesting that sediment particles may well follow the Reynolds-averaged flow velocity less gravitational settling. Hence, we adopt an Equilibrium Eulerian approximation [Ferry and Balachandar, 2001], where the particle velocity can be written as an expansion of fluid velocity and sediment properties. Here, only the leading order term is considered in the expansion [e.g., Cantero et al., 2008; Hsu et al., 2009; Torres-Freyermuth and Hsu, 2010] and particle velocity is calculated by fluid velocity plus particle settling velocity. This approximated relationship of particle velocity is further substituted into the Eulerian-Eulerian two-phase equations for particle-laden flow to obtain the present governing equation appropriate for fine sediment (see more detailed discussions by Torres-Freyermuth and Hsu [2010]). The continuity equation is written as

equation image

where u and w are the flow velocity in the streamwise (x) direction and vertical (z) direction and ϕ is the sediment volumetric concentration. The resulting flow momentum equations in the x- and z- directions are given by:

equation image

and

equation image

where p is the fluid pressure, g = 9.8 m/s2 is the gravitational acceleration, ρ is the fresh water density, s = ρs is the specific gravity of sediment, and Δρ represents the density change due to salt concentration ϕs:

equation image

with α1 = 0.824, α2 = −0.1809 and α3 = 0.483. The flow momentum transport is affected by both fluid stresses τf, including viscous and turbulent stresses, and sediment stresses. Sediment stresses, which are incorporated in the original equations presented by Torres-Freyermuth and Hsu [2010], are only important when the sediment concentration is large. This work considers a maximum sediment concentration of only 36 g/L, which corresponds to volumetric concentration of 0.0136, so the sediment stresses are ignored. In the present formulation, the gravitational terms in the z-momentum equation allow both sediment-induced and salt-induced buoyancy-driven flow to be calculated.

[10] Sediment volumetric concentration is calculated by mass conservation:

equation image

where the first term on the right hand side is the gravitational settling with sediment settling velocity represented by Ws, and the second and third terms are the sub-grid and turbulent diffusion with the sub-grid diffusion coefficient of sediment Dc, the turbulent eddy viscosity νt and the turbulent Schmidt number σc. The sediment settling velocity is calculated by the Stokes law:

equation image

[11] Salt concentration is also calculated by mass conservation:

equation image

where Ds = 1.6 × 10−9 m2/s is the molecular diffusion coefficient of salt and σs is the corresponding turbulent Schmidt number. For dilute flow considered here, the sub-grid diffusion coefficient of sediment Dc is set to be equal to the molecular viscosity of the fluid (Dc = ν = 1.0 × 10−6 m2/s) for simplicity. Because we shall investigate convective settling in turbulent flow where instability is mainly driven by gravitational settling [Hoyal et al., 1999a] and/or interfacial mixing [Maxworthy, 1999; McCool and Parsons, 2004], the effect of Dc is not investigated. Numerical experiments also suggest varying Dc by a factor 10 does not change the qualitative features of the model results. However, turbulent Schmidt numbers (σs and σc) are somewhat sensitive to model results. They are part of the closure coefficients in turbulent flow and the choice of their values are discussed in the next section.

2.1.2. Turbulence Closure

[12] Solutions of the flow mass and momentum (2)(4), the sediment volumetric concentration (6), and the salt concentration (8) equations require closure for flow turbulence and the associated transport. In this study, turbulent Reynolds stresses are calculated by the eddy viscosity assumption employing a two-equation turbulence closure model. The total fluid stress is calculated as:

equation image

where i, j = 1,2 and δij is the Kronecker delta. Turbulent eddy viscosity, νt, is related to the fluid turbulence kinetic energy, k, and turbulent dissipation rate, ɛ, by

equation image

[13] The turbulent kinetic energy, k, and dissipation rate, ɛ, are calculated by their balance equations [e.g., Torres-Freyermuth and Hsu, 2010; Hsu et al., 2009]. The k equation is written as

equation image

and the ɛ equation is written as

equation image

[14] Standard k-ɛ model coefficients (Cμ = 0.09, C = 1.44, C = 1.92, σk = 1.0 and σɛ = 1.3) are used in this study [e.g., Rodi, 1987]. Empirical coefficient C*3ɛ for the density stratification term in the ɛ equation has several values that have been suggested [e.g., Rodi, 1987; Umlauf and Burchard, 2005; Choi and García, 2002]. In this study, we follow Umlauf and Burchard [2005] and use C*3ɛ = 0.0 for stable density stratification and C*3ɛ = 1.0 when stratification becomes unstable. Following Torres-Freyermuth and Hsu [2010], turbulent Schmidt numbers (σc and σs) are both set to be 0.5, and C3ɛ is set to be 1.0 for typical dilute sediment-laden flow. C*3ɛ is the empirical coefficient most sensitive to model results, followed by σc, σs and C3ɛ. This sensitivity is one of the weak points in the two-equation turbulent closure of particle-laden flow. However, this set of empirical coefficients falls well within the range documented in the existing literature and gives reasonably good results when compared with laboratory data on turbidity and density currents (see section 2.3).

2.1.3. Boundary Conditions

[15] The model forcing is based on prescribed inlet conditions at the left boundary of the domain that creates an inflow over a prescribed vertical range. Inflow conditions used to drive the model are horizontal velocity, sediment volumetric concentration, and salt concentration. Sediment in the inflow is characterized by grain diameter and density. The bottom boundary is modeled as a simple wall. Standard rough-wall logarithmic law is imposed as the boundary condition for velocity parallel to a solid boundary. For more detailed discussions on bottom boundary condition for flow velocity, turbulence kinetic energy and turbulent dissipation rate, readers are referred to Torres-Freyermuth and Hsu [2010]. There are several choices of sediment resuspension flux in the numerical model [see e.g., Hsu and Liu, 2004; Hsu et al., 2009]. Resuspension of sediment from the bed is set to be zero and only deposition is allowed in this study. Since there is no wave forcing, the top boundary is modeled as a free-slip surface with the gradients of streamwise velocity, sediment concentration and salt concentration set to be zero, i.e., equation image. The right side of the domain, where the flow exits, is modeled with an open boundary condition. To account for the free surface pressure, the top-right point of the boundary is prescribed with a fixed pressure p = 0. Everywhere else along the entire boundary of the computational domain, the Neumann boundary condition is applied.

2.2. Numerical Implementation

[16] The proposed mathematical formulation for salt-stratified sediment-laden flow is implemented into a two-dimensional vertical (2DV) numerical model solving Reynolds-averaged Navier-Stokes equations with a k-ɛ turbulence closure called COBRAS (Cornell Breaking Wave and Structure) [Lin and Liu, 1998]. The backbone of the COBRAS model is the numerical model RIPPLE developed originally by Los Alamos National Laboratory [Kothe et al., 1991] as a two-dimensional Navier-Stokes solver with a volume of fluid (VOF) scheme for free surface tracking. Partial-cell treatment [Kothe et al., 1991] is utilized for solid obstacle in the computational domain (if any). This code has been validated and utilized extensively to study various surface wave and sediment transport problems [e.g., Lin and Liu, 1998; Hsu and Liu, 2004; Torres-Freyermuth and Hsu, 2010]. However, it is extended to a salt-stratified flow and density current here. This work, though not utilizing the free surface tracking capability of the model, looks to further validate the model in this area and to further utilize the numerical model as a tool to study convective sedimentation.

2.3. Model Validation

[17] Before using the model to study convective sedimentation, it is important to test its ability to model density and turbidity currents. This is done by comparison to laboratory experiments reported by García [1993].

2.3.1. Experiment

[18] Laboratory experiments reported by García [1993] are statistically 2-D experiments involving turbidity and density currents traveling over a slope transition. García [1993] conducts experiments with both sediment and saline driven currents and, in many cases, these currents experience a hydraulic jump at the slope transition. For turbidity current runs, many experimental runs are conducted with variations to sediment size, sediment density, and inlet velocity. Also conducted are saline current experiments, set up with similar nondimensional parameters to the turbidity current runs. For model validation, two of these experiments (NOVA7, SAL29) of similar fractional density are first presented in order to test the model capability in calculating both the fine sediment transport and the salinity transport. Then, model validation is presented with three runs in DAPER series of slightly coarser sediment.

2.3.2. Results

[19] The turbidity run (NOVA7) is driven by a particle-laden current of Novaculite sediment (s = 2.65 and D = 4μm) with volumetric concentration Φ0 = 0.0073. On the other hand, the SAL29 run is driven by saline current with similar fractional density (Δ=0.012) to that of NOVA7. In both cases, turbidity/saline current of velocity U0 = 11 cm/s is sent through an inlet of height h0 = 3 cm into ambient fresh water. The temperature difference for these runs is fairly small (no more than 1°c) and, as such, is not considered in the numerical model. In Figure 1, a snapshot of numerical model results for turbidity current (Figure 1a) and saline current (Figure 1b) runs at t = 450 s are presented. The flow structures for turbidity current and saline current runs are visualized through sediment concentration and salinity contours, respectively. The mass concentration c is calculated from volumetric concentration via c = ρsϕ. The flow structures observed in the experiment are reproduced well by the numerical model. The internal hydraulic jump, where the density current changes from upstream super-critical to downstream sub-critical flow, is clearly present at the point of slope change. Moreover, because these two density currents have the same averaged fractional density and the sediment settling velocity in the turbidity current run is very small (Ws = 0.02 mm/s, deposition is negligible within the test section of the flume, see García [1993] for more details), we observe quite similar flow behavior between these two runs, consistent with laboratory observation by García [1993].

Figure 1.

A snapshot of 2DV numerical model results for (a) sediment concentration (g/L) from flow conditions of NOVA7 run of García [1993], and (b) salinity (ppt) for flow condition of SAL29 run of García [1993] at t = 450 s. The inlet is located at the left boundary from z = 0.40∼0.43 m. For Figure 1a, the flow from the inlet is of fresh water with sediment concentration 19.3g/L. For Figure 1b, the flow from the inlet is of saline water with salinity 14.8 ppt. In both runs, the inlet flow velocity is 11.0 cm/s and receiving (ambient) water is fresh (zero salinity).

[20] To carry out more detailed validation, model predicted time-averaged velocity profiles are further plotted against measured data in Figure 2 in the super-critical regime (x = 3 m) and sub-critical regime (x = 8 m). It is clear that the model is capable of predicting velocity associated with the density currents. Specifically, the numerical model captures the variation of current height, the magnitude of current velocity as well as the location of the peak velocity for both the turbidity current run (Figure 2, left) and the saline current run (Figure 2, right). Numerical experiments with different grid resolution suggest while the upstream supercritical flow can be well-predicted by fine vertical resolution (Δz = 2 mm), reasonably high streamwise resolution (Δx = 2cm) is also needed to predict the downstream sub-critical flow because of the rapid flow evolution during the slope transition (i.e., internal hydraulic jump).

Figure 2.

Comparison of averaged current velocity for (left) NOVA7 and (right) SAL29 between measured data reported by García [1993] (symbols) and model results (curves). Averaged velocity profiles are taken at 300 and 800cm downstream from the inlet, before (supercritical) and after (subcritical) the hydraulic jump, respectively. Model results are averaged over a time period of 400 s (t = 100 to 500 s) for sediment current and 300 s (t = 100 to 400 s) for saline current.

[21] It is also worth mentioning that when the sediment-induced density stratification terms in the k and ɛ equations are turned off (the second to the last terms in equations (11) and (12)), predicted sediment/salt becomes well-mixed in the water column and density current structures disappear. This suggests sediment/salt induced density stratification plays a critical role in damping the carrier flow turbulence, suppressing turbulent mixing and allowing the density anomaly to be confined near the bed to drive gravity flow. On the other hand, the viscous drag term in the k and ɛ equations is only of minor importance in changing the observed flow features due to relatively dilute concentration considered in this study. These observations are consistent with prior numerical modeling studies [e.g., Choi and García, 2002; Cantero et al., 2008]. For the same reason, model results are sensitive to the closure coefficient associated with the density stratification in the ɛ-equation (C*3ɛ). The value used here for stable stratification (C*3ɛ =0) gives the best results when compared with measured data. Using C*3ɛ = 0 for stable density stratification is also consistent with other modeling work for density current [e.g., Rodi, 1987; Umlauf and Burchard, 2005]. More detailed discussions on the sensitivity of closure coefficients are presented by Snyder [2010].

[22] We have also carried out model-data comparison with DAPER series (DAPER1, DAPER2 and DAPER6). All the DAPER series runs are of grain diameter D = 9μm, specific gravity s = 2.65 and inlet velocity of U0 = 8.3 cm/s. However, each run is of different inlet sediment concentration of Φ0 = 0.00143, 0.00183, 0.00372 for DAPER1, DAPER2 and DAPER6, respectively. Following García [1993], results for the DAPER series presented here are normalized by the layer thickness h of the turbidity current, the layer-averaged velocity U and the layer-averaged sediment concentration C [see García, 1993, equations (9) and (10)]. Results shown in Figure 3 suggest the numerical model is able to predict the flow velocity and sediment concentration profiles that agree well with the measured data in both the upstream supercritical and downstream subcritical regimes.

Figure 3.

Comparison of normalized (a) sediment concentration profiles and (b) velocity profiles for DAPER1, DAPER2 and DAPER6 runs between measured data reported by García [1993] (symbols) and model results (curves). Model results are averaged over a time period of 300 s (t = 200 to 500 s). Results are normalized by layer thickness h and layer averaged sediment concentration C and flow velocity U following García [1993].

[23] From these model-data comparisons, we conclude that the numerical model is capable of predicting turbidity and saline currents. This gives us confidence in further utilizing the numerical model to study problems related to sediment-laden salt-stratified buoyancy driven flow. Specifically, in this study we utilize the numerical model to investigate the dynamics of convective sedimentation from a hypopycnal plume.

3. Convective Sedimentation

3.1. Idealization and Relevant Nondimensional Parameters

[24] In this study, we investigate the one of the highly probable mechanisms associated with the rapid deposition of sediment from hypopycnal plumes [e.g., Warrick et al., 2004] caused by turbulent convective instabilities, known as sediment fingers or convective sedimentation [Maxworthy, 1999; Parsons et al., 2001; McCool and Parsons, 2004]. The numerical model presented in the previous sections is utilized. A schematic of the numerical model domain setup for hypopycnal plume runs is illustrated in Figure 4. The domain initially consists of stationary salt-water with salt concentration Φs = 0.035 (i.e., salinity 35 ppt). An inlet is located at the upper left of the domain, which allows a prescribed inflow of sediment-laden fresh water (salinity 0 ppt). Each run studied here is characterized by a prescribed inlet flow velocity U0, inlet height h0, inlet sediment concentration Φ0, and sediment settling velocity Ws (see Table 1). We further define the density of the ambient saltwater as ρa = ρ + Δρ, and the density of the sediment-laden plume as ρm = ρ + Φ0(ρs − ρ).

Figure 4.

Schematic representation of domain setup for numerical model runs of idealized hypopycnal plume flows.

Table 1. Dimensional and Nondimensional Parameters of the Different Numerical Model Runs
CaseD (μm)Ws (mm/s)C0 (g/L)U0 (m/s)h0 (m)ReFrγRepΔ (mm)
1a20.00360.66250.10.415415000.3010.01467.2 × 10−6∼0
1b20.00362.120.10.415415000.3060.04677.2 × 10−6∼0
1c20.00366.6250.10.415415000.3240.1467.2 × 10−63.42 × 10−7
1c_120.00366.6250.20.415830000.6480.1467.2 × 10−61.15 × 10−5
1c_220.00366.6250.050.415207500.1620.1467.2 × 10−65.44 × 10−8
1c_320.00366.6250.20.831660000.4580.1467.2 × 10−68.08 × 10−5
1d20.003613.250.10.415415000.3560.2927.2 × 10−64.67 × 10−7
1e20.003626.500.10.415415000.4640.5847.2 × 10−65.32 × 10−4
1f20.003636.000.10.415415000.6600.7947.2 × 10−61.1 × 10−3
2a100.08980.66250.10.415415000.3010.01468.98 × 10−43.84 × 10−7
2b100.08982.1200.10.415415000.3060.04678.98 × 10−47.24 × 10−8
2c100.08986.6250.10.415415000.3240.1468.98 × 10−47.70 × 10−7
2d100.089813.250.10.415415000.3560.2928.98 × 10−41.24 × 10−4
2e100.089826.500.10.415415000.4640.5848.98 × 10−40.013
2f100.089836.000.10.415415000.6600.7948.98 × 10−40.038
3a200.360.66250.10.415415000.3010.01467.2 × 10−33.46 × 10−5
3b200.362.120.10.415415000.3060.04677.2 × 10−31.49 × 10−5
3c200.366.6250.10.415415000.3240.1467.2 × 10−31.14 × 10−6
3c_1200.366.6250.20.415830000.6480.1467.2 × 10−31.15 × 10−3
3c_2200.366.6250.050.415207500.1620.1467.2 × 10−32.53 × 10−5
3c_3200.366.6250.20.831660000.4580.1467.2 × 10−39.44 × 10−3
3d200.3613.250.10.415415000.3560.2927.2 × 10−31.59 × 10−4
3e200.3626.500.10.415415000.4640.5847.2 × 10−30.0327
3f200.3636.000.10.415415000.6600.7947.2 × 10−30.1244
4a401.4370.66260.10.415415000.3010.01460.05752.69 × 10−5
4b401.4372.120.10.415415000.3060.04670.05755.49 × 10−4
4c401.4376.6250.10.415415000.3240.1460.05751.73 × 10−4
4d401.43713.250.10.415415000.3560.2920.05756.25 × 10−3
4e401.43726.500.10.415415000.4640.5840.05750.4894
4f401.43736.000.10.415415000.6600.7940.05750.8379
5a603.230.66260.10.415415000.3010.01460.1940.0105
5b603.232.120.10.415415000.3060.04670.1940.0718
5c603.236.6250.10.415415000.3240.1460.1940.256
5c_1603.236.6250.20.415830000.6480.1460.1940.293
5c_2603.236.6250.050.415207500.1620.1460.1940.322
5c_3603.236.6250.20.831660000.4580.1460.1940.447
5d603.2313.250.10.415415000.3560.2920.1940.670
5e603.2326.500.10.415415000.4640.5840.1943.006
5f603.2336.000.10.415415000.6600.7940.1948.652
6a805.750.66260.10.415415000.3010.01460.460.1492
6b805.752.120.10.415415000.3060.04670.460.4566
6c805.756.6250.10.415415000.3240.1460.461.7298
6d805.7513.250.10.415415000.3560.2920.464.3317
6e805.7526.500.10.415415000.4640.5840.468.5775

[25] In order to generalize our idealized study, it is useful to present model results via nondimensional parameters. Simple dimensional analysis suggests four nondimensional parameters control this phenomenon. The first two nondimensional parameters, the inlet Reynolds number

equation image

and the inlet (internal) Froude number

equation image

essentially control the dynamics of the positively buoyant plume and the associated interfacial waves [e.g., Geyer and Smith, 1987]. The inlet Froude number controls the large-scale plume dynamics and, in the present idealized simulations, its effect can be represented by the inlet Reynolds number and the density ratio. Hence, its further use here is unnecessary. The next two parameters, the density ratio,

equation image

and the particle Reynolds number (nondimensionalized settling velocity)

equation image

control sedimentation. They are known to be critical parameters controlling convective sedimentation in still water [Hoyal et al., 1999a; Hogg et al., 1999]. These nondimensional parameters allow us to compare results between different sets of model runs to determine the relative importance of these flow parameters to the development of convective fingers and rapid sedimentation. With these nondimensional parameters, it is also possible to generalize our results to larger scale.

[26] All the computational runs presented in this study have the same domain length L = 5 m and domain height H = 1.2 m. The grid resolution is set to be Δx = 0.01 m and Δz = 0.005 m to ensure that convective fingers can be resolved. The bottom boundary is implemented as a rough bed. Resuspension of sediment from the bed is disabled but sediments are allowed to settle into the bed. These settled sediments are recorded to obtain the total deposition (see section 3.2.3). Existing laboratory evidence suggests that the largest convective finger occurs via interfacial mixing induced convective instability [Maxworthy, 1999] which is only of a scale (width) of several centimeters. Therefore, to ensure high numerical resolution to resolve the fingers, the highest inlet Reynolds number investigated here is Re = 1.66 × 105. With such fine spatial resolution, we cannot afford at the same time to carry out field-scale computations (Re = 106∼107). Nevertheless, the inlet Reynolds number in the present numerical study is larger than all the existing laboratory investigations (Re≈104).

[27] According to the initial salinity of Φs = 35 ppt and the grain density is specified to be ρs = 2650 kg/m3, the critical concentration for hyperpycnal flow (i.e., Δρm = Δρa) is 45.37 g/L. In this study, we focus on hypopycnal flow and the maximum inlet sediment concentration studied is C0 = 36 g/L (Φ=0.0136). A wide range of settling velocity (Ws = 0.036 mm/s to 5.75 mm/s) is tested via changing the grain size from D = 2 μm to D = 80 μm. We note here that in realistic sediment-laden plume, flocculation must occur and the resulting sediment settling velocity depends on flocculation [Dyer, 1989]. Because it remains highly empirical to model floc dynamics in sediment-lade turbulent flow [e.g., Son and Hsu, 2011], flocculation is not explicitly considered in this study. Nevertheless, we attempt to better understand convective sedimentation by testing a range of settling velocities on the order of O(0.1∼1) mm/s, which covers typical values of settling flocs. Most simulations are carried out for a fixed set of inlet velocity and inlet height. Hence, the inlet Reynolds number remains constant, allowing us to mainly study the effects of inlet sediment concentration and settling velocity (variations in γ and Rep). In some cases, we also vary the inlet velocity and inlet height (hence inlet Re) in order to evaluate how the change in stratified shear flow can affect the convective sedimentation. A summary of flow parameters carried out in this study is presented in Table 1.

3.2. Model Results

3.2.1. Flow Regimes

[28] Numerical model results suggest the inlet sediment concentration is the most critical parameter controlling the resulting flow features, followed by the sediment settling velocity. Qualitatively, four flow regimes can be identified. The main features of each flow regime are discussed in this subsection. More quantitative analyses are presented in sections 3.2.2 to 3.2.4.

[29] When the inlet sediment concentration is greater than about 20 g/l (γ > ≈0.5), convective instability occurs immediately after the turbid front enters the domain and a large convective finger/plume is formed (Figure 5a). This convective plume immediately descends to the bottom, bringing a significant amount of sediment and fresh water to the bottom, and becomes an underflow that continues to propagate downstream (Figure 5b). This lower plume can cause significant deposition near the inlet. Significant reduction of salinity near the bed is also observed (in Figure 5b the salinity drops from 35 ppt to about 29 ppt). Meanwhile, the surface plume continues to propagate downstream as more convective fingers occur (t > 20 s, not shown here). However, these subsequent fingers appear to be less intense than the first one. The observed flow features are similar to the divergent plumes reported by prior laboratory experiments [McCool and Parsons, 2004] and field study of Warrick et al. [2004, 2008]. In this study, we shall also characterize this flow as a divergent plumes (DP) regime (Regime 1).

Figure 5.

Snapshots of numerical model results representative for Regime 1, i.e., the divergent plume (DP) regime (C = 36 g/l, run 4f) at (a) t = 9 s and (b) t = 20 s. Figures 5a (top) and 5b (top) show color plot of sediment concentration and Figures 5a (bottom) and 5b (bottom) show salinity.

[30] The observed Regime 2 is defined here as the intense convective fingers (ICF) regime (see Figure 6). This regime occurs generally for runs with settling velocity greater than O(1) mm/s (Rep > 0.1) or sediment concentration between O(1) g/l and 20 g/l (0.2 < γ < 0.5). Multiple convective sediment fingers are developed in Regime 2. These fingers trigger rapid sediment deposition to the bed and also bring freshwater downward to the middle of the water column. Figure 6a illustrates a more detailed flow structure of the ICF regime. Convective fingers emerge soon after the plume front enters the domain (the first finger usually occurs within t = 30 s). However, the sediment concentration in these fingers is significantly lower than the first large finger in the DP regime and hence underflows are not initiated as sediment fingers approach the bottom. Moreover, no noticeable reduction of salinity is observed near the bottom. These convective fingers predicted by the numerical model are similar to those observed in prior laboratory experiments. Specifically, the development of these fingers is asymmetric with respect to the density interface, resembling a typical settling-induced or interfacial mixing induced instability [e.g., Hoyal et al., 1999a; Parsons et al., 2001; McCool and Parsons, 2004]. The predicted finger size is about a few centimeters. The descending speed of the finger is around 1∼3 cm per second, which brings both sediment and fresh water more rapidly (than the Stokes settling velocity) toward the bottom. However, as these fingers descend, they quickly lose fresher water due to positive buoyancy and turbulent mixing. Eventually, these fingers only contain sediment and the salinity inside the fingers becomes similar to that of the ambient flow. For example, in Figure 6a (top), the finger occurs around x = 1 m identified by sediment concentration penetrating to about z = 0.4 m, while in Figure 6a (bottom) the same finger identified by salinity only penetrates to about z = 0.6 m. Despite few main differences between the DP regime and the ICF regime discussed here, from fluid mechanics perspective, we believe the divergent plume observe in Regime 1 can be considered as a very intense version of the fingers observed in Regime 2.

Figure 6.

Snapshot of numerical model results for run 5c (inlet sediment concentration C = 6.625 g/l and grain diameter D = 60 μm) at (a) t = 55 s and (b) t = 75 s.

[31] The observed Regime 3 is defined here as the weak convective finger (WCF) regime (see Figure 7). This regime occurs generally for runs with settling velocity smaller than O(1) mm/s (Rep < 0.1) and sediment concentration below 10 g/l (γ < 0.2). The WCF regime can also be considered as the transitional condition between ICF regime and Regime 4, i.e., the negligible convective finger (NCF) regime, where development of fingers is either not observed or extremely weak such that no deposition is detected. The parametric boundaries of WCF and NCF appear to depend on both the inlet sediment concentration and the sediment settling velocity to a similar degree and hence shall be defined more clearly in section 3.2.4. Figure 7 presents a snapshot of a typical WCF regime. The settling velocity in this case is 9 times smaller than that shown in Figure 6 although they have the same inlet sediment concentration. In this case, similar convective fingers are observed but the sediment concentration in these fingers is about 10 times smaller than that in the ICF regime shown in Figure 6 (notice the color plot range in Figure 6b is set to be 10 times smaller than that in Figure 6a). Consequently, the downward momentum of the sediment finger in the WCF regime is much weaker. In Figure 7, it can be clearly observed that freshwater is usually not contained in the sediment fingers after a very short distance away from density interface. These fingers descend more slowly and are easily advected by the ambient shear flow. Many of these fingers cannot reach the bottom and it may take several fingers to merge into one larger finger to have sufficient downward momentum to reach the bed (not shown). The dynamics of the convective finger and the resulting sediment deposition are clearly more complicated because ambient shear flow and turbulence also play major roles in determining the fate of these fingers.

Figure 7.

Snapshots of numerical model results for run 3c of smaller grain size (C = 6.625 g/l and D = 20 μm) at (a) t = 60 s and (b) t = 85 s. Sediment concentration in the convective fingers for this case is much lower than the case shown in Figure 6 (notice the color plot range is set much smaller in this figure).

[32] In summary, it is not straightforward to simply characterize convective instability and the resulting finger development and sediment deposition for a hypopycnal plume using a single flow feature and flow parameter. The characterization of the convective sedimentation with four flow regimes allows us to further quantify convective sedimentation more effectively. In the following section, we utilize two different physical quantities that can be calculated from the numerical model results, namely, the time scale to instability and the total deposit, in order to characterize the convective sedimentation with the nondimensional parameters discussed previously.

3.2.2. Timescale to Instability

[33] Because the occurrence of convective fingers is a type of instability, it is common to characterize instability with a timescale (i.e., inverse of a growth rate) or a length scale for the instability to occur. Using their measured data and an analogy to thermal convection, Hoyal et al. [1999a] suggest a critical length scale δcrit for the convective instability to occur. This critical condition is defined by a Grashof (Gr) number:

equation image

where the reduced gravitational acceleration is defined as g′ = Φ0(ρsρ) g/ρa and κ is the thermal diffusion coefficient. Using a thermal analogy with a critical Grashof number of 1000, a critical length scale for convective instability to occur can be written as δcrit ∼ (1000ν2/g′)1/3. Hoyal et al. [1999a] argue that once particle settling from the upper layer has moved downward to this critical level, convective instabilities are able to form. In addition to finding δcrit, they also observe the interfacial layer thickness (δcrit), finger wavelength (λ), and finger thickness (l) scale proportionately with each other as (δcritλ∼5l). Because it is more difficult to evaluate δcrit in our study due to the presence of ambient shear flow, we evaluate the timescale Tcrit for the instability to occur. Qualitatively, this timescale Tcrit can be related to δcrit by sediment settling velocity as Tcrit = δcrit/Ws.

[34] Examining the numerical results of each run, we can define and evaluate T1 as the time required for the first mature finger to develop a length of 20 cm (measured from the fresh-saltwater interface). Due to the effect of ambient shear flow and the development of interfacial waves/instabilities, it is somewhat objective to define this interface. Furthermore, in such a complicated flow condition, it is perhaps more meaningful to seek a timescale that involves the development of many fingers. Hence, we also evaluate another timescale T2 defined as the time required for the first five fingers to reach z = 30 cm above the bed. The 30 cm is chosen to be close enough to the bed but not too close such that finger descending speed begins to be affected by the bottom solid boundary. For the following discussion on T1 and T2, the divergent plume regime (Regime 1) is excluded because a large convective plume always appears immediately after the computation.

[35] For Regimes 2–4, there appears to be a general trend (Figure 8) that the timescale for instability (both T1 and T2) become smaller for larger inlet sediment concentration and larger sediment settling velocity. This general trend is consistent with prior laboratory study for convective fingers in still water [Hoyal et al., 1999a]. However, it is also clear that when inlet sediment concentration becomes larger, the dependence on settling velocity is weaker. It can be also observed that the dependence of T1 on settling velocity is not always monotonic (i.e., sometimes sediment of larger settling velocity has larger T1). As we shall discuss in more detail later, this can be mainly attributed to the finger development being further complicated by interfacial waves and the level of ambient turbulence.

Figure 8.

Timescale to instability, i.e., (a) T1 and (b) T2 (sec) plotted against inlet sediment mass concentration C0 (g/L) for six different sediment settling velocities.

[36] On the other hand, the dependence of T2 on settling velocity and concentration is less noisy. This is partly because T2 represents an averaged behavior of the first five convective fingers. When we examine T2 more carefully, it is clear that as sediment settling velocity gets larger, the dependence on inlet concentration gets weaker (and vice versa). It is worth mentioning that for the runs of the two smallest settling velocities (0.0036 mm/s and 0.0898 mm/s) with inlet sediment concentration around or below about 2 g/L, there is no more than three fingers developed within the 300 s computation. Hence, a thick dashed line is indicated in Figure 8b to make a distinction (these cases are shown as a representative value T2 = 310 s above the thick-dashed line). In each of these runs, the total sediment deposit is almost zero and is considered in Regime 4 (NCF regime). Another feature of T2 that shall be pointed out here is that for the two intermediate settling velocity runs (0.359 mm/s and 1.437 mm/s) with the smallest inlet sediment concentration (0.63 g/L), the associated T2 is smaller (less stable) than that of the next higher inlet concentration runs. This feature is again due to more complex interaction between convective finger and ambient stratified shear flow in the WCF regime that shall be elaborated in section 3.2.4.

3.2.3. Total Deposits

[37] As discussed in the previous section, the upward erosion flux is set to zero in this study. However, sediment settling flux D = ϕWs at the first grid point above the bottom boundary is time-integrated throughout the entire computation. Hence, we can further calculate the spatially averaged (averaged over the entire domain length L) total sediment deposit height as:

equation image

where C* = 0.6 is an estimate of closely packed bed concentration and Γ = 300 s is the time period for the computation. Notice that in the present simulations, the distance between the bottom of the plume and the bed is no more than 0.8 m. The timescale for sediments to settle within the computational domain can be calculated from the domain length (L = 5 m) divided by the typical plume velocity U = 0.1 m/s, which gives 25 s. If sediment settles only by Stokes law, where the largest values tested here is 5.75 mm/s, the largest settling distance is only about 29 cm. In other words, if sediments merely settle with Stokes law without convective instability, no deposit will be detected within the computational domain.

[38] Since the main purpose of studying convective sedimentation is to better understand the resulting rapid sediment deposition, we shall present nondimensionalized results with the total deposit height Δ normalized by grain diameter d. Figure 9 presents model results of normalized total deposit height plotted against particle Reynolds number (nondimensionalized settling velocity) for different inlet density ratio (inlet sediment concentration). It is evident that for the two largest inlet concentration runs (Φ = 36 and 26.5 g/l, or γ > ∼0.5) in the divergent plume (DP) regime (see Figure 5), the deposits are at least several factors (for large Rep) to several order of magnitudes (for small Rep) larger than those in the convective finger regimes. For cases with particle Reynolds number greater than about 0.1, which we identify as the ICF regime, the normalized total deposit height is directly proportional to γ and Rep. This observation is consistent with laboratory experiments of convective sedimentation in still water: that the convective finger development is qualitatively proportional to the magnitude of sediment concentration and particle settling velocity [e.g., Hoyal et al., 1999a]. This suggests that once a convective finger is developed in the ICF regime, the downward momentum of the finger is sufficiently strong to overcome the ambient shear flow and the associated mixing.

Figure 9.

Normalized total deposit height plotted against particle Reynolds number for different inlet density ratio. The dashed line represents a normalized total deposits height of 6 × 10−6, a very small number such that Δ/d smaller than that number is considered to have negligible deposit.

[39] For run 1a and run 1b (with the smallest Rep and the two smallest γ) the normalized total deposit height is smaller than 6 × 10−6. These two cases are considered to have almost negligible deposit and are categorized into the NCF regime. The cases that lie in between the ICF regime and the NCF regime are considered to be in the WCF regime. Normalized total deposit height in the WCF regime falls within the range of 10−5 < Δ/d < 0.1. Unlike the ICF regime, the amount of deposit in the WCF regime does not seem to have a monotonic dependence on γ (inlet sediment concentration) and settling velocity (Rep). For example, run 3c and run 3a (or 3b) both have the same Rep. However, run 3c has a smaller deposition height even though the inlet sediment concentration in this case is in fact 10 times larger than that in run 3a. A similar situation can be observed between runs 2a and 2b, and runs 4b and 4c. What we observed here is essentially a main feature of the WCF regime: that the convective fingers are weak and hence are easily affected by the ambient mixing processes.

[40] The conditions of the ambient stratified shear flow for cases 3a and 3c are demonstrated in Figure 10 based on spatially averaged streamwise velocity equation imageu〉, density anomaly 〈Δρ〉/ρ (due to both salinity and sediment), and turbulent intensity 〈It〉 =〈equation image〉, where “〈 〉 ” denotes the spatial-average operator in the x-direction over the entire domain length L. In both cases (as well as other cases), a pronounced two-layer shear flow system is established with a high-velocity low-density layer on top of a slow-velocity high-density layer. Strong shear at the density interface is expected to generate turbulence. However, turbulence and mixing in the lower layer may be suppressed due to stable density stratification. According to Figure 10 (middle), it is clear that the stable density stratification for case 3c is weaker because higher inlet sediment concentration (10 times higher than that in case 3a) adds a negative density anomaly. Such weaker stable density stratification gives stronger turbulent intensity in the lower layer for case 3c (see Figure 10, right). Because convective fingers in the WCF regime are weak, as they penetrate through the water column in case 3c the stronger turbulence causes more intense turbulent suspension, which significantly delays settling of the convective finger. In summary, although case 3c has larger inlet sediment concentration, the resulting turbulent intensity in the lower layer becomes stronger and hence sediment settling and total deposit are reduced. It is also worth mentioning that the gradient Richardson number at the stratified shear layer shown in Figure 10 for case 3a is around 0.8 while the gradient Richardson number for case 3c is only around 0.3, suggesting that for case 3c the turbulence and mixing processes in the lower layer are expected to be more intense.

Figure 10.

Spatially averaged (over the entire domain length L) streamwise (left) velocity profiles, (middle) density anomaly and (right) turbulent intensity for case 3a (red-dashed curves) and case 3c (black-solid curves).

3.2.4. Discussions

[41] In section 3.2.1, four different flow regimes are identified and their main features are discussed (see Figures 5, 6, and 7). From the most intense convective sedimentation to negligible sedimentation, these flow regimes are identified as (1) the Divergent Plume (DP) regime, (2) the Intense Convective Finger (ICF) regime, (3) the Weak Convective Finger (WCF) regime, and (4) the Negligible Convective Finger (NCF) regime. Based on 35 cases with different combination of inlet sediment concentration and settling velocity, main characteristics of these convective sedimentation regimes are discussed through the timescale for the instability to occur and total deposit height (section 3.2.2 and 3.2.3). The existence of these flow regimes in the two-dimensional parametric space of particle Reynolds number Rep (nondimensional settling velocity) and density ratio γ (inlet sediment concentration) for the given inlet Reynolds number Re = 41500 are illustrated in Figure 11.

Figure 11.

The existence of the flow regimes of convective sedimentation in a 2D parametric space of density ratio γ and particle Reynolds number Rep for the given inlet Reynolds number runs of Re = 4.15 × 104.

[42] DP regime only occurs when γ is larger than certain critical value (in this case it is about 0.5). ICF regime occurs at intermediate value of γ (0.2 < γ < 0.5) for all range of Rep or at all range of γ when Rep is large (Rep > 0.1). Both DP and ICF regimes can have large and comparable sediment deposits. Hence, high sediment concentration is expected to be measured near the bottom for these two regimes. However, the main difference according to the present model results is that a noticeable decrease of salinity can be observed only for the DP regime.

[43] For smaller γ and Rep values (0.05 < γ < 0.2, 10−4 < Rep < 0.1), the downward momentum of the convective finger becomes weaker and therefore the fate of the finger is significantly affected by the ambient shear flow and turbulent mixing. The resulting sediment deposition is much less than that in the DP and ICF regimes. Moreover, larger inlet concentration does not necessarily produce more deposits in the near-field because higher inlet concentration adds a negative buoyancy anomaly to the upper layer and allows more turbulent mixing in the lower layer, which enhances turbulent suspension in the lower layer and discourages deposition. Finally, the NCF regime is characterized by sediments of very small settling velocity and inlet sediment concentration (Rep < 10−4 and γ < 0.05). The convective finger development is very slow and weak and results in virtually no sediment deposition within the range and duration of computation.

[44] Numerical investigation presented so far is based on a fixed inlet Reynolds number of Re = 41500. Additional runs (see Table 1) for different inlet Reynolds number are discussed here by halving or doubling the inlet velocity and/or inlet height for a fixed inlet concentration of C = 6.625 (γ = 0.146) with three different particle Reynolds number (Rep = 7.2 × 10−6, 7.2 × 10−3, 0.194). With a small particle Reynolds number (Rep ≪ 0.1) and inlet Re = 41500, convective sedimentation is in the WCF regime. Further increasing the inlet Reynolds number enhances the convective sedimentation and total deposit (see two smaller Rep cases in Figure 12). Numerical results here are consistent with the laboratory study of McCool and Parsons [2004], which concludes that increasing inlet flow velocity (equivalent to inlet Re) enhances turbulent-mixing induced convective sedimentation. However, model results here further indicate that for cases of large particle Reynolds number (Rep > 0.1) where convective sedimentations are already in the ICF regime, increasing the inlet Reynolds number only marginally affects the resulting deposit (see crosses in Figure 12). This feature is consistent with the characteristics of the ICF regime: that convective fingers are sufficiently strong and are relatively unaffected by the ambient shear flow and turbulent mixing.

Figure 12.

Normalized deposit height plotted against different inlet Reynolds number for three different particle Reynolds number (Ws = 0.0036, 0.36, 3.234 mm/s) and fixed density ratio of γ = 0.146 (C = 6.625 g/l).

[45] Notice that for the series of runs shown in Figure 12 with Rep = 7.2 × 10−3 (circles), the case with inlet Re = 41500 has a smaller deposit than that of Re = 20750. Again, the main characteristic of WCF regime discussed in Figure 10 is also demonstrated here. As the inlet Reynolds number is decreased from Re = 41500 to Re = 20750 by reducing the inlet velocity 50% (compare case 3c and case 3c_2 in Table 1), the intensity of the stratified shear flow and hence the mixing below the surface plume are also reduced. The reduced turbulence in the lower layer allows more sediment to settle in the case of a lower inlet Reynolds number.

[46] Using the small-scale numerical model presented here, a preliminary guideline for the parameterization of convective sedimentation can be referred to the parametric map shown in Figure 11. For instance, in practical coastal modeling of initial deposition, if the resulting nondimensional parameters Rep and γ in the river plume are in the DP or ICF regime, it may then be justified to specify an artificially large settling velocity on the order of cm/s in the coastal models as a simple parameterization of rapid settling due to convective sedimentation. We also note that because numerical model results presented here are of smaller Reynolds number than that in the field condition, according to Figure 12, the guideline presented in this study is a conservative estimate on the occurrence of convective sedimentation.

4. Conclusion

[47] A two-dimensional-vertical (2DV) numerical model is developed to study fine sediment transport in salt-stratified turbulent flow. The numerical model, solving the non-hydrostatic Reynolds-averaged Navier-Stokes equations with a k-ɛ turbulence closure, is validated with laboratory experiments on turbidity and salinity currents [García, 1993]. The numerical model is then utilized to study convective sedimentation in idealized conditions. Based on 35 numerical model runs for different inlet sediment concentration (density ratio) and settling velocity (particle Reynolds number) at a fixed inlet Reynolds number of Re = 41500, four flow regimes are identified. For a large density ratio, a divergent plume (DP regime) is developed immediately after the turbid front enters the domain. The lower plume then propagates as an underflow with relatively low salinity near the bottom, causing high deposition. For an intermediate density ratio and large particle Reynolds number, intense convective fingers (ICF regime) occur which are only marginally affected by ambient stratified shear flow. In both aforementioned regimes, large sediment deposits result with the magnitude of the deposit proportional to inlet sediment concentration and settling velocity. Further reducing the density ratio or particle Reynolds number gives weak convective fingers (WCF regime) that are significantly affected by the ambient flow, and the resulting deposition is also weak. Additional numerical model runs with different inlet velocity/height (i.e., inlet Reynolds number) suggest that total deposits are unaffected by inlet Reynolds number when convective sedimentation is sufficiently intense (i.e., in the DP and ICF regimes). However, convective sedimentation in the WCF regime can be enhanced when inlet Reynolds number is increased, which is consistent with prior laboratory studies [e.g., Maxworthy, 1999; McCool and Parsons, 2004] for interfacial mixing induced convective sedimentation.

[48] Our idealized numerical experiments presented here can provide, at least qualitatively, the criterion for the occurrence of convective sedimentation as well as guidance on which flow regime can be expected. However, more future studies both in terms of theoretical/numerical modeling and observations are necessary. Large exports of terrestrial sediments during river flooding often coincide with storm events where surface waves are also intense. However, the role of surface waves to enhance or suppress the occurrence of convective sedimentation remains unclear. Floc aggregation and breakup occur concurrently with sedimentation. In other words, the settling velocity of sediment is a variable during convective sedimentation but this process is not explicitly simulated in this study. Finally, more field evidences of convective sedimentation are certainly necessary to understand the main cause of rapid deposition at a river mouth during flooding of small mountainous river.

Acknowledgments

[49] This study is supported by National Science Foundation (OCE-0913283; OCE-0926974) and Office of Naval Research (N00014-11-1-0176). This study has also benefited from many useful discussions with Rocky Geyer, Jim Kirby, and Xiao Yu.