Convective instability in sedimentation: Linear stability analysis

Authors

  • Xiao Yu,

    Corresponding author
    1. Center for Applied Coastal Research, Civil and Environmental Engineering, University of Delaware, Newark, Delaware, USA
    • Corresponding author: X. Yu, Center for Applied Coastal Research, Civil and Environmental Engineering, University of Delaware, Newark, DE 19716, USA. (yuxiao@udel.edu)

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  • Tian-Jian Hsu,

    1. Center for Applied Coastal Research, Civil and Environmental Engineering, University of Delaware, Newark, Delaware, USA
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  • S. Balachandar

    1. Mechanical and Aerospace Engineering, University of Florida, Gainesville, Florida, USA
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ABSTRACT

[1] Convective sedimentation in a stably stratified saltwater is studied using the linear stability analysis. Convective sedimentation is known to occur due to the double-diffusive mechanism and the settling-driven mechanism. In this study, a semi-empirical closure of sediment diffusivity based on the long-range hydrodynamics effect is adopted. The sediment phase can act as either the slow- or fast-diffusing agent in the double-diffusive system for the given salt diffusivity. Moreover, the settling-driven effect is proportional to the square of the sediment diameter via Stoke settling law. We consider sediment concentration (grain size) in the upper freshwater layer to be in the range of 0.1 to 39.4 g/l (2 to 60 µm), which is on top of a saltwater layer with salinity 35 ppt. Linear stability analysis allows us to identify the dominant mechanism that triggers the instability, the growth rate, and the resulting characteristic finger width. Model results suggest that for fine sediment with grain diameter smaller than 10 µm (settling velocity 0.09 mm/s), double-diffusive mechanism controls the instability and the resulting sediment finger size is of millimeter scale. For the given threshold of growth rate of O(0.01) s−1, the minimum sediment concentration is about 8–15 g/l. For grain size greater than or around 10 µm, the settling-driven mechanism dominates and instabilities occur at sediment concentration as low as O(0.1) g/l with centimeter-scale fingers. Our findings may contribute to a better understanding on the observed rapid sediment removal in the plume of small mountainous rivers.

1 Introduction

[2] Sedimentation in a stratified fluid is a common occurrence in geophysical and industrial flow systems. In many of these applications, including the initial deposition of terrestrial sediments from river plumes [Warrick et al., 2008; Parsons et al., 2001], fall layers of volcanic ash (tephra) in the deep sea [Carey, 1997], vertical transport of plankton [Green and Diez, 1995], flotation processes [Fessas and Weiland, 1982], and solidification processes [Sokolowski and Glicksman, 1992], rapid particle settling with an effective settling velocity significantly greater than the Stokes settling velocity of a single particle is often observed. The mechanisms responsible for such rapid settling are usually attributed to convective instabilities, or convective sedimentation [Blanchette et al., 2005; Voltz et al., 2001; Burns and Meiburg, 2012].

[3] In estuaries and river mouths, interactions between the terrestrial fine sediment and the density-stratified ambient flow induced by salinity and temperature are of crucial importance in determining the removal rate of sediment from the river plume and the location of the initial deposition [Warrick et al., 2008; Geyer et al., 2004; Hill et al., 2000]. Because the density of seawater is greater than that of the fresh water, when a river plume loaded with fine sediment enters the sea, it often stays above the seawater (as in a hypopycnal flow) provided that the sediment concentration remains dilute [Wright and Nittrouer, 1995]. During episodic flooding events of many small mountainous rivers [Milliman and Syvitski, 1992; Mulder and Syvitski, 1995], river-borne sediment concentration may exceed 30 m 40 g/l and the highly turbid river plumes may descend down to the sea bottom and propagate as underflows (i.e., hyperpycnal flow) causing large deposits near the river mouth [Wright et al., 1990; Milliman et al., 2007]. For more commonly observed hypopycnal plumes, the location of the initial deposition is mainly determined by the effective settling velocity of sediment either as primary particles or as floc aggregates [Hill et al., 2000]. Due to the fine-grained nature of these terrestrial sediments, their still-water terminal settling velocity as given by Stokes drag law is typically no more than 0.01–1 mm/s. With such a small settling velocity, the predicted location of deposits is often quite far away from the river mouth, which is inconsistent with many field observations [Warrick et al., 2008; Geyer et al., 2004]. Laboratory experiments suggest when the sediment concentration exceeds O(1) g/l, convective instabilities commonly develop at the density interface and form finger-like convection in the lower layer, which leads to a significantly enhanced effective settling velocity of sediment [Parsons et al., 2001; McCool et al., 2004; Maxworthy, 1999]. Recently, these laboratory observations are also reproduced reasonably well by a numerical study based on a Reynolds-averaged approach for salt-stratified particle-laden flow [Snyder and Hsu, 2011].

[4] In the absence of ambient shear and large-scale convection due to a mean flow, there are several mechanisms that can contribute to the occurrence of the convective sedimentation [Green, 1987; Hoyal, 1999a; Huppert et al., 1991]. Two major sources of instabilities are the double-diffusive convection and the settling-driven convection [Hoyal, 1999a]. Several laboratory experiments have been carried out to quantify the sediment fluxes driven by double-diffusive mechanism [Green, 1987; Parsons and Garcia, 2000; Hoyal et al., 1999b; Chen, 1997]. The double-diffusive instability occurs because of the difference in the diffusion coefficients between the two density altering agents in the carrier fluid (salt and sediment in this study). The resulting convective sediment finger size is typically of millimeter scale. In comparison, only a handful of studies [Hoyal, 1999a; Blanchette et al., 2005; Snyder and Hsu, 2011] focused on the settling-driven convective sedimentation. In a stratified ambient, due to the settling of particles, an unstable bulk density profile can develop over time, which in turn can undergo Rayleigh-Taylor instability at the downward propagating particle front [Voltz et al., 2001; Bush et al., 2003]. In this context, it has been observed that even for sediment concentration as low as a few gram per liter, centimeter-sized convective sediment fingers appear and the instability significantly contributes to the enhanced effective vertical migration and the apparent settling velocity of the sediment [Hoyal, 1999a; Blanchette et al., 2005].

[5] Recent work by Burns and Meiburg [2012] reported a linear stability analysis of convective sedimentation in a salt-stratified condition. They identified the key parameter as the ratio of two length scales, the settling distance of sediment particles, and the diffusive spreading distance of salt. They also found that the increase of the ratio of the unstable layer thickness to the diffusive interface thickness of the salt concentration profile tends to significantly increase the growth rate and changes the dominant mode from double-diffusive to Rayleigh-Taylor instability mode. Burns and Meiburg [2012] set the sediment diffusivity to a very small constant. Hence, the sediment phase always plays the role of the slow-diffusing agent in the double-diffusive system. However, for sediment particles in natural river plumes, the sediment particle size can range from several micrometers (~1 µm) to several tens of micrometer (~100 µm) while the sediment concentration can be significantly greater than O(0.1) g/l. In such condition, the long-range hydrodynamic effect of inter-particle interaction becomes dominant [Guazzelli and Hinch, 2011]. Segrè et al. [2001] gives a semi-empirical formula for the effective sediment diffusivity due to long-range interaction above the dilute limit of 0.01% [Segrè et al., 1997], which depends on the sediment size and sediment concentration. According to the semi-empirical formulation, fine sediment in freshwater above a saltwater can play either as the slow- or fast-diffusing agent, depending primarily on the sediment size. Because both the sediment diffusivity and settling velocity are mainly determined by the sediment size, it is crucial to study the convective sedimentation with both the double-diffusive instability and the settling-driven instability considered. In this study, we investigate the convective sedimentation with a linear stability analysis. Following Burns and Meiburg [2012], our work presented here further expands their study with a more realistic description of sediment diffusivity in order to understand the conditions, under which the double-diffusive and settling-induced instabilities become significant for a wide range of grain size and sediment concentration commonly encountered in the estuaries and river mouths. Specifically, with a given threshold for growth rate, we focus on identifying the minimum concentration for the occurrence of convective sedimentation and the expected convective finger size.

2 Model

2.1 Physical Problem

[6] We consider sediment particles settling in a two-layer quiescent stratified fluid. The carrier fluid is water with kinematic viscosity ν and density ρ0. Initially, the upper layer is fresh water that carries sediment particles at a given concentration, while the lower layer is salt water without sediment particles. The flow examined in this study is shown schematically in Figure 1. A Cartesian coordinate system is used, and it extends from z = − H to z = H in the vertical direction with H as the half water column depth. The flow is at rest initially. Because of the diffusion processes and the settling of sediment particles, the base state develops in time. Error functions are used to prescribe the base flow sediment concentration and salt concentration profiles for simplicity. Error function is the solution of the laminar diffusion equation with a constant diffusion coefficient starting from a discontinuous initial condition [Fischer et al., 1979] and serves to provide a good physical approximation to the actual sediment and salt concentration profiles. The initial salt and sediment concentration profiles are given by

display math(1)
display math(2)

where math formula and math formula are the characteristic length scales controlling the steepness of salt concentration and sediment concentration profiles, respectively. They are also the characteristic length scales of the diffusion processes. ΔΦs and ΔΦ are the absolute values of the difference in salt and sediment concentration between the top and bottom boundaries and ε measures the separation between the centers of the two profiles.

Figure 1.

Parameters and base density variation profiles used in the present linear stability analysis.

[7] For river plumes in nature, ambient shear and strong turbulent mixing typically co-exist with the two-layer system described above. Incorporating ambient shear flow and the associated shear instabilities makes the present problem too difficult to tackle with a linear stability analysis. As a first step, we focus on investigating the competing effects of the double-diffusive and settling-driven mechanisms in the development of convective instabilities, and the effect of ambient shear is neglected in the present study.

[8] In this study, the initial salt concentration in the lower layer away from the density interface is set to 0.035 (salinity 35 ppt), which gives ΔΦs = 0.035 and the corresponding density is 1.027 × 103 kg/m3 (by equation (10)). A series of analyses with different sediment concentration in the upper layer (ΔΦ) are carried out to investigate the dynamics of this system. The sediment grain size determines both the Stokes settling velocity and the sediment diffusivity. A wide range of sediment diameters is studied to investigate the relative importance of the double-diffusive and settling-induced mechanisms. The half depth of the water column H affects the dynamics of system through the boundary effects. Provided that H is sufficiently large, the effect of H is negligible in all the simulations. In this study, H is set to be 10 cm, and our model experiments suggest that this choice of H is sufficiently large for the math formula, math formula, and ε values we choose, such that the calculated growth rate does not change with H. As we will discuss later, the magnitudes of math formula and math formula affect the growth rate of instabilities. For instance, smaller math formula gives a shaper initial concentration profile, which leads to a larger growth rate. In this study, a small fixed value of math formula is chosen for the initial salt concentration profile. We then proceed with our investigation by setting math formula and ε = 0 as an idealized baseline scenario (sections 4.1 and 4.2). A more realistic scenario where the ratio of math formula scales with their corresponding diffusion coefficients is then investigated in section 4.3. To better incorporate the effect of gravitational settling before the appearance of large-scale motions in the present linear stability analysis, we also vary ε, which is expected to increase with time as the sediment particles settle (section 4.4).

2.2 Governing Equations

[9] To model sedimentation in a stably stratified ambient, such as that in a hypopycnal flow, the sediment concentration is typically dilute (much less than 40 g/l, or volumetric concentration φ < 1.5%) and the sediment particles (clay and silt) are usually fine with grain diameters (d) in the range of 1 to 60 µm and the specific gravity γ = 2.65. A dilute suspension of mono-dispersed fine spherical particles is considered in this study. Both the fluid and sediment phases are treated as continuum, and accordingly, the Eulerian-Eulerian two-fluid framework is adopted. In the standard Eulerian-Eulerian approach, the sediment velocity field must be solved with a set of momentum equations that are coupled with the fluid phase through inter-phase momentum exchange terms. The resulting two-way coupled governing equations are complicated and computationally very expensive.

[10] For sediment considered in this study, the Stokes settling velocity of single particles is quite small, in the range of 1 × 10− 6 to 3.2 × 10− 3 m/s. The particle response time, defined as math formula, is in the range of 1 × 10− 7 to 5.3 × 10− 4 s. We estimate the time scale of convective motion resulted from the instability to be larger than the particle response time. In the limit of small particle Stokes number (math formula; ratio of particle-to-fluid time scale), we simplify the momentum equations for the sediment phase by adopting the Equilibrium Eulerian approach [Ferry and Balachandar, 2000; Balachandar and Eaton, 2009]. In this case, the sediment velocity v can be explicitly given as the sum of carrier fluid velocity u, Stokes settling velocity, and an O(St) correction that accounts for the leading order inertial behavior of the sediment particles [Balachandar and Eaton, 2009].

[11] We apply the Boussinesq approximation, which is appropriate for the dilute condition, and the fluid momentum equation can be written as [Cantero et al., 2007]

display math(3)

where u is the velocity of the carrier fluid, ρ is the bulk fluid density which is a function of sediment concentration φ and salt concentration φs ∗, and math formula is the unit vector (0,0,1). For the sediment phase, the governing equation of mass conservation is given as

display math(4)

where κ is the diffusion coefficient of the sediment phase and v is the sediment phase velocity. In the Equilibrium Eulerian approximation, the sediment phase velocity can be algebraically related to the local fluid velocity and its gradients, which will be discussed later.

[12] Because the intensity of convective instabilities strongly depends on the double-diffusive mechanism [Green, 1987; Chen, 1997; Parsons and Garcia, 2000], the diffusion coefficient of sediment phase needs to be determined. For very fine particles with sub-micron size, Brownian motion determines the magnitude of the effective diffusion coefficient [see Coussot and Ancey, 1999]. In this study, we focus on sediment particles that are several micrometers in diameter. Velocity fluctuations due to the presence of neighboring particles (the so-called hydrodynamic dispersion effect) dominate over the Brownian motion [Ham and Homsy, 1988]. In general, shear-induced dispersion of particles is a main mechanism of particle transport [Stickel and Powell, 2005]. However, in this study we carry our linear stability analysis of convective sedimentation in an initially quiescent fluid without ambient shear and the shear-induced diffusivity is neglected. The magnitude of the particle diffusion coefficient, therefore, depends on the settling velocity, sediment diameter, concentration, and concentration gradient of the sediment phase [Peter et al., 2004; Mucha and Brener, 2003; Martin et al., 1995; Guazzelli and Hinch, 2011]. In the present linear stability analysis, the correction due to concentration gradient on sediment diffusivity is neglected for simplicity [Tee et al., 2002] and the diffusion coefficient of the sediment phase is assumed to be isotropic, which is given as [Segrè et al., 1997; Segrè et al., 2001]

display math(5)

where ΔV is the velocity fluctuations in the direction parallel to gravitational acceleration, and ξ is the mean free path of diffusion, which is given by experimental data [Segrè et al., 1997] as ξ = 5.5dφ∗ − 1/3. The velocity fluctuations in the direction parallel to the gravitational acceleration is given by [Segrè et al., 2001]

display math(6)

where math formula and math formula is the Stokes settling velocity of an isolated sediment particle in a fluid of dynamic viscosity η0. η(φ) is the effective viscosity of fluid with the presence of sediment. The structure factor can be given as S(φ,0) = [∂ Π/∂ φ]− 1, with Π given by the Carnahan-Starling equation of state Π(φ) = (φ + φ∗ 2 + φ∗ 3 − φ∗ 4)/(1 − φ)3 [Russel et al., 1989]. The same empirical relationships as Segrè et al. [2001] for effective viscosity is adopted here: η(φ)/η0 = (1 − φ/0.71)− 2. The resulting diffusivity of the sediment phase used here can be written as

display math(7)

where math formula and math formula. For very dilute flow with φ = O(1), it can be shown that the sediment diffusivity is only weakly dependent on the sediment concentration. Compared with the Stokes settling velocity, which scales with d∗ 2, the sediment diffusivity shown in equation (7) is a fast-growing function of sediment diameter d (scales with d∗ 3). With a fixed value of salt diffusivity in water and the range of sediment diameters considered in this study, the sediment phase can act as either a slow-diffusing agent or a fast-diffusing agent in the double-diffusive system.

[13] The transport equation of salt concentration φs ∗ is given as

display math(8)

where κs = 1.3 × 10− 9m2/s is the diffusion coefficient of salt in water. The bulk fluid density can now be expressed as a function of the sediment and salt concentration

display math(9)

where α and β describe variations of density around a reference state due to sediment and salt, respectively, as

display math(10)

where γ = ρs/ρ0 = 2.65 and β ≈ 0.745 [Perianez, 2005].

2.3 Dimensionless Equations

[14] To include both the double-diffusive and the settling-driven mechanisms, the characteristic velocity and length scales are chosen as U = (νg)1/3 and L = ν/U = (ν2/g)1/3. The non-dimensionalized governing equations then can be written as

display math(11)
display math(12)
display math(13)

with the corresponding dimensionless variables defined as

display math

where ΔΦ and ΔΦs are used to non-dimensionalize the sediment concentration and salt concentration. Four dimensionless parameters can be defined here

display math(14)

[15] The Schmidt number Scs and Scp are the ratio of kinematic viscosity (momentum diffusivity) to the diffusion coefficients of salt in water and sediment in water, respectively. Ra and Ras are Rayleigh numbers associated with sediment and salt. In this study, we focus on the variation of sediment concentration for a given Ras value (i.e., salinity). A sediment-to-salt ratio Λ can be defined as the ratio of the two Rayleigh numbers,

display math(15)

and Λ becomes the key parameter controlling the dynamics of the system. The ratio Λ is also a widely used parameter in other particle-laden stratified flow problems [Sparks et al., 1993; Burns and Meiburg, 2012].

[16] With the equilibrium Eulerian approximation, the sediment phase velocity can be expressed algebraically as an expansion in Stokes number, which can be truncated at O(St) for St ≪ 1. This approach has been tested [Ferry and Balachandar, 2000] in a variety of particle-laden turbulent flows and is shown to be fairly accurate provided that St < 1.0. In this study, fine sediment with diameter in the range of 2 to 60 µm is studied. The corresponding Stokes number St can be shown to be between 1 × 10− 4 and 0.2. Therefore, the equilibrium Eulerian approach can be adopted where sediment particles follow the local carrier fluid velocity with corrections due to particle settling and inertial effect. The sediment velocity v can then be expressed as [Ferry and Balachandar, 2000; Balachandar and Eaton, 2009; Druzhinin, 1995]

display math(16)

where b = 3/(2γ + 1) redefines the particle-to-fluid density ratio, math formula is the non-dimensionalized settling velocity, and ut is the material derivative of fluid velocity. The above expression is accurate to O(St) and higher-order terms can also be derived [Ferry and Balachandar, 2000]. Through utilizing equation (16), two more non-dimensional parameters are introduced in the present problem, that is, the non-dimensionalized settling velocity V0 and the Stokes number St.

[17] Periodic boundary conditions are enforced in horizontal directions. The boundaries of the system are the top and bottom boundaries, where no-slip and no-penetration conditions are enforced to the carrier fluid velocity. For non-dimensional sediment concentration, we apply

display math(17)

at the boundaries and thus conserve the mass of sediments within the system. For salinity, no-flux boundary condition math formula is applied at both the top and bottom walls to guarantee the mass conservation.

3 Linear Stability Analysis

[18] One of the most common instabilities induced by density variations in fluids is the Rayleigh-Taylor instability. When the heavier fluid is on top of the lighter fluid, the system tends to minimize its potential energy and instabilities may occur at the density interface. For initially stable density stratification with lighter sediment-water mixture on top of denser salt water, the unstable bulk density profile will form as the sediment particles settle through the density interface [Hoyal, 1999a]. The ratio Λ, which measures the relative buoyancy effects due to sediment and salt, is of the most importance. Additionally, the sediment diffusion coefficient predicted by equation (7) can be either smaller or greater than that of salt depending on the grain diameter d and sediment concentration. If the ratio between the fast-diffusing agent and the slow-diffusing agent is sufficiently large, the double-diffusive mechanism is expected to contribute significantly to the development of instabilities. We present a linear stability analysis applicable to this system, in which sediment particles settle with the non-dimensional settling velocity V0 in a stably stratified ambient induced by salt.

3.1 Governing Equation of Perturbations

[19] Initially, the flow is at rest with mean sediment concentration profile and salt concentration profile specified a priori. The perturbations are added to the main flow, so that

display math(18)
display math(19)
display math(20)
display math(21)

where u′, v′, and w′ are velocity perturbations in the horizontal, vertical, and spanwise directions, respectively, and p′, φ′, and φs ′ are perturbations associated with pressure, sediment concentration, and salt concentration. Linearizing the governing equations (equations (11) to (13)), we obtain the governing equation for perturbations of vertical velocity and salt concentration as

display math(22)
display math(23)

where math formula. For the governing equation of sediment concentration perturbation, we apply Taylor's expansion of f(φ) and df(φ)/dz to linearize the governing equation and obtain

display math(24)

where f0, f1, and f2 are functions of the mean concentration Φ0, with f0 and f1 as the linear approximation of f and df/dz, respectively:

display math(25)
display math(26)
math image(27)

[20] The base is translationally invariant along the horizontal x-y plane, and as a result, we can simplify above governing equations to a 2D problem. To investigate the stability of this system, we use the method of normal modes. We introduce a stream function of the velocity perturbations written as math formula, where ω is the frequency and k is the wave number. In this study, we investigate the temporal instability of the system with ω defined as a complex variable but keeping the wave number k real. By definition, the perturbed velocities can be given by the stream function as

display math(28)

and the incompressibility condition math formula is automatically satisfied. After eliminating the pressure terms and substituting the normal modes into above equations, a system of ordinary differential equations is obtained:

display math(29)
display math(30)
math image(31)

[21] The corresponding boundary conditions are

display math(32)

and

display math(33)

3.2 Special Case: Double-Diffusive System When V0 = 0

[22] If we neglect the effect of gravitational settling (i.e., set V0 = 0), the above system converges to the typical double-diffusive system, which has been widely studied in oceanography for salt finger phenomenon [Stern, 1960; Schmitt, 1994], and analytic solutions can be given with linear distribution of both salt and sediment concentration. For V0 = 0, the marginal stability with ω = 0 gives the governing equation of perturbations

display math(34)

[23] Here the effective Rayleigh number can be defined as

display math(35)

[24] In the present problem of convective sedimentation in hypopycnal flow stated in Figure 1, math formula is negative and math formula is positive and Ras > Ra is required to have an initially stable density stratification. If the ratio math formula is less than the diffusivity ratio Scs/Scp, we obtain Raeff > 0. For the instability to occur, there is a minimum value of Raeff which depends on the boundary conditions and is in general of order 103 [Howard, 1964]. It is clear that as Ra or the sediment concentration in the upper layer becomes large, the effective Raeff increases and the system becomes more unstable. The analogy with the Rayleigh's problem also suggests that the formation of statically unstable bulk density profile does not necessarily lead to large-scale convection because of the presence of viscous effect [Schmitt, 1994].

[25] When Raeff > Racrit, instabilities will develop. The classic model of the thermohaline convection [Stern, 1960] can be used to determine the horizontal wave number kc at which the instability grows the fastest and the growth rate of the most unstable mode. This specifies the horizontal wavelength of the observed finger patterns and the initial horizontal finger width. There has been experiments [Stern, 1960; Baines and Gill, 1969] showing that the dominant mode is usually on the order of several millimeter for double-diffusive convection with constant gradient of both salinity and temperature.

3.3 Numerical Method

[26] With the gravitational settling term in the governing equations, no analytic solutions can be found. The above ODE system is numerically solved by Chebyshev-Tau method. The density interface is located at z = 0 initially. However, the grid points are clustered at z = ± 1 with a Chebyshev collocation method. To locally refine the grid at the density interface (z = 0), the adaptive rational collocation method [Tee and Trefethen, 2006; Cueto-Felgueroso and Juanes, 2009] is applied in this study. Perturbations are expanded in the barycentric form, which provides a general framework that includes both polynomial and rational approximations. A rational function which interpolates a grid function math formula at points z0, z1, …, zN can be expressed as

display math(36)

where w1, w2, …, wN are barycentric weights. In particular, the above expression is a polynomial that interpolates at the Chebyshev points {zk = cos(/N), k = 0, …, N} for

display math

[27] A conformal mapping, which transforms the Chebyshev grid into a grid with adaptively clustering points near steep gradients of the solution, is implemented to smoothly track the steep gradients. After discretizing the above equations, a generalized eigenvalue system math formula is obtained, where math formula is a vector containing math formula, math formula, and φs. To find a nontrivial solution, we seek for the eigenvalues of the above equation. The dimensional growth rate G (s− 1) is defined as the negative of the imaginary part of ω divided by the characteristic time scale T = L/U (i.e., math formula), with G > 0 meaning the system is unstable.

4 Model Results

[28] The settling velocity of sediment particles, calculated by the Stokes law, is proportional to the square of the sediment diameter. The downward propagation of the initial sediment concentration profile promotes the formation of an unstable density profile, which may lead to instabilities. The sediment diffusivity, which is of crucial importance in a double-diffusive system, is proportional to the cube of the sediment diameter as shown in equation (7). In reality, the double-diffusive and the settling-driven convective instabilities will occur concurrently. It is, therefore, critical to understand under which circumstance both the double-diffusive convection and the settling-driven convection are of comparable importance, as well as when either the settling-driven or the double-diffusive convection dominates. In this study, the salt water density is fixed, and we vary the sediment concentration in the upper layer. To investigate the convective sedimentation relevant to typical sediment-laden river plume in estuarine and coastal environments, the range of sediment concentration investigated in this study is between 0.4 and 39.4 g/l (ΔΦ = 1.5 × 10− 3 to 1.5% or Λ from 10− 2 to 1) and the range of grain size d is between 2 and 60 µm. Because the salt water density is fixed (ΔΦs = 0.035) in our study, we only show our result as a function of the sediment-to-salt ratio Λ. Model results are first presented with a baseline scenario in which δp = δs and ε = 0. More realistic scenarios with the ratio of δp to δs dictated by the respective diffusion coefficients and with nonzero values of ε will be discussed later.

4.1 Concentration Dependence of Sediment Diffusivity and the Effect of the Inertial Term

[29] Sediment concentration in a hypopycnal river plume is typically dilute with volumetric concentration of no more than 1.5% (at Λ = 1). It is thus of interest to evaluate how important is the sediment concentration variation in the sediment diffusivity given by equation (7) and in the stability analysis. In other words, we want to investigate the importance of including f(φ) and its derivatives in the stability analysis. If the effect of sediment concentration on sediment diffusivity can be established to be small, then we can take f(φ) = 1 and greatly simplify the stability equation (31). Figure 2a shows the result of dimensional growth rate G as a function of ratio Λ for d = 2 µm with δp = δs = H/20 and ε = 0. Notice that the results shown in Figure 2a with or without considering the dependence of sediment diffusivity on sediment concentration are almost identical, suggesting that we can take the sediment diffusivity κ to be independent of sediment concentration without changing the characteristics of the system. In the rest of the paper, we simply use κ0 as sediment diffusivity and sediment diffusivity becomes only a function of sediment diameter.

Figure 2.

(a) The dependence of growth rate G on Λ for d = 2 µm. The effect of concentration dependence on sediment diffusivity is negligible in this study. (b) The dependence of growth rate G on Λ for d = 60 µm with Scp = 0.7, Scs = 700 and ΔΦs = 0.035. The effect of the inertia term in the equilibrium Eulerian approach is negligible in this study.

[30] Similarly, we also examine the importance of the particle inertia. According to the equilibrium Eulerian approximation, the first-order term associated with the particle inertia effect is proportional to the fluid acceleration and the particle Stokes number (the third term on the right-hand side of equation (16)). The largest Stokes number studied in this paper is from the largest particle with diameter d = 60 µm (i.e., typical silt sediment) and the corresponding Stokes number around 0.24. Figure 2b shows the comparison of dimensional growth rate G for a range of Λ by including and excluding the inertial term for d = 60 µm. The growth rates calculated with and without the particle inertia term are very close. Therefore, it can be concluded that the effect of particle inertia on the instability is negligible. The main reasons for the negligible particle inertia effects are due to the small Stokes number and the small fluid acceleration during the initial linear instability development. Therefore, in the rest of the paper, the sediment velocity will be taken to be a simple sum of the local fluid velocity and the settling velocity, where only the first two terms on the right-hand side of equation (16) are retained. It is, however, noted here that if a fully nonlinear 3D simulation were carried out, the particle inertia effect might be more pronounced.

4.2 Convective Sedimentation With ε = 0, δp = δs = H/20

[31] When ε = 0 and δp = δs, the system is initially stably stratified. The vertical gradient of the bulk density is negative dρ/dz < 0 in the entire water column. As sediment particles start to settle downward across the density interface due to the nonzero settling velocity, an unstable density profile is established, which may lead to Rayleigh-Taylor instability [Blanchette et al., 2005; Voltz et al., 2001]. Meanwhile, the double-diffusive mechanism co-exists during the settling process, which can also enhance the particle migration and trigger instabilities. In the limit of very fine sediment particles with very small settling velocity (V0 ∼ 0), the double-diffusive mechanism can be of crucial importance [Parsons and Garcia, 2000; Hoyal et al., 1999b; Chen, 1997]. Since the growth rate and characteristic finger size are distinctly different for double-diffusive and Rayleigh-Taylor instabilities, it is crucial to identify the relative importance of these two mechanisms in this baseline scenario. As we discuss in the next section, depending on the sediment diameter, sediment phase can play either as the slow-diffusing agent or the fast-diffusing agent in a double-diffusive system and the resulting growth rate of convective instability is not a monotonic function of sediment diameter.

4.2.1 Slow-Diffusing Sediment

[32] For sediment particles with diameter d = 2 µm, the non-dimensional settling velocity is very small V0 = 1.68 × 10− 4. From equation (7), we obtain Scp = 18900 and Scs = 700. The ratio of the diffusion coefficients between the fast-diffusing agent (salt) and the slow-diffusing agent (sediment phase) in this case is κs/κ0 = 27. The double-diffusive mechanism may trigger instability for large effective Rayleigh number. Figure 3a shows the calculated growth rate G as a function of ratio Λ (solid curve). The growth rate of the fastest growing mode increases rapidly with increasing Λ, that is, with the initial sediment concentration in the upper layer. The growth rate is also calculated with Stokes settling velocity V0 intentionally set to zero in Figure 3a (dashed curve). These two curves almost overlap with each other, suggesting that for very fine particles the settling-driven mechanism is very weak and the double-diffusive mechanism dominates the initial development of instabilities. The growth rate with Stokes settling included is in fact slightly smaller than that without settling. This damping effect could be due to the smearing of the vorticity field discussed in Burns and Meiburg [2012]. Burns and Meiburg [2012] found this damping effect to scale with the settling velocity and correspondingly the growth rate to be inversely proportional to the settling velocity. In this case, a critical value of Λcrit for the instability to occur clearly exists. This is similar to many salt finger studies where a critical Rayleigh number is often used [Schmitt, 1994]. In practical applications, what is of main concern is often a timescale for the large-scale motion to appear. This timescale is associated with the inverse of a threshold value of the growth rate. A more detailed discussion on the threshold growth rate for the present application is discussed later.

Figure 3.

Growth rate as function of Λ by including and excluding the gravitational settling term. (a) d = 2 µm, Scs = 700, Scp = 18900, and V0 = 1.68 × 10− 5. (b) d = 20 µm, Scs = 700, Scp = 20, and V0 = 1.68 × 10− 2. (c) d = 60 µm, Scs = 700, Scp = 0.7, and V0 = 0.151.

[33] Figure 4a shows the growth rate G as a function of wave number k for the sediment-to-salt ratio Λ = 0.5 and 0.75, respectively. The growth rate G is not a monotonic function of wave number k, and there exists a single peak, which corresponds to the most unstable mode. It can be observed that the wave number of the most unstable mode only weakly depends on the ratio Λ for very fine sediments, where instability is mainly controlled by the double-diffusive mechanism. The observed convective finger size (width) can be qualitatively estimated using the most unstable mode given by the linear stability analysis. In this case, the predicted sediment finger size is in the order of millimeter (with most unstable mode around 90 cm− 1). Although the effect is quite small, model results indicate that finger size is slightly larger for smaller Λ. When the settling velocity is very small, there is sufficient time for the double-diffusive instability to grow and the instability is mainly caused by the double-diffusive mechanism. Our finding here is also consistent with the earlier work by Burns and Meiburg [2012] for small settling velocity. When the double-diffusive mechanism is the dominant mechanism causing instabilities, the characteristic finger size is in the order of millimeters. In oceanography, the field-observed double-diffusive salt finger size [Stern, 1960] is often in the order of millimeters.

Figure 4.

Growth rate as function of wave number k for different ratio Λ = 0.5 and 0.75, respectively. (a) d = 2 µm, Scs = 700, Scp = 18900, and V0 = 1.68 × 10− 4. (b) d = 20 µm, Scs = 700, Scp = 20, and V0 = 1.68 × 10− 2. (c) d = 60 µm, Scs = 700, Scp = 0.7, and V0 = 0.151.

4.2.2 Fast-Diffusing Sediment

[34] For sediment diameter with d = 60 µm, the non-dimensional settling velocity is V0 = 1.51 × 10− 1, which is significantly larger than that of d = 2 µm discussed in previous section. For this grain size, we obtain Scp = 0.7 and Scs = 700. The ratio of the diffusion coefficient between the fast-diffusing agent (sediment phase) and the slow-diffusing agent (salt) is κ0/κs = 1000. The sediment phase acts as the fast-diffusing agent in the double-diffusive system, and the double-diffusive mechanism is expected to contribute to the development of instabilities. Meanwhile, the settling velocity is also quite large, and the settling-driven mechanism is also expected to play an important role. Figure 3c shows the growth rate as a function of Λ (solid curve). By comparing to the growth rate calculated with settling velocity intentionally set to zero (dashed curve), the importance of settling in determining the resulting growth rate can be ascertained. When the settling velocity is large, the front of the sediment concentration profile propagates rapidly such that the double-diffusive mode does not have enough time to grow and its contribution to the instability development is of less importance. In comparison with the case of fine sediment d = 2 µm, the growth rate caused by the settling-driven mechanism is significantly larger than that by the double-diffusive mechanism. Figure 4c shows the growth rate as a function of wave number k. The most unstable mode predicted by the linear stability analysis is around 6 cm− 1. This gives the characteristic finger size on the order of centimeter, which is much larger than the finger size shown in Figure 4a for the very fine sediment particle. Laboratory experiment [Hoyal, 1999a] also indicates larger sediment finger when settling-driven mechanism is the major cause of the instability. Figure 4c also suggests that the characteristic finger size increases slightly when Λ decreases.

[35] For sediment with diameter d = 20 µm, the non-dimensional settling velocity is V0 = 1.68 × 10− 2 and we have Scs = 700 and Scp = 20. In this case, although the sediment phase still acts as fast-diffusing agent, the ratio of diffusion coefficients between the fast-diffusing agent (sediment phase) and the slow-diffusing agent (salt) is only κ0/κs = 35, which is much smaller than that of the case with d = 60 µm. The effect of double-diffusive mechanism is therefore expected to be weak. Figure 3b shows the comparison of growth rate with and without the Stokes settling term. It is clear that when the settling velocity is set to zero, the double-diffusive mechanism predicts almost no occurrence of instability. Hence, the observed instability is almost completely due to the settling-driven mechanism. Figure 4b shows the corresponding growth rate for d = 20 µm as a function of wave number k at two different values of Λ. It is clear that the wave number k of the most unstable mode depends more strongly on the ratio Λ when the settling-induced mechanism dominates the dynamics of the system. The wave number k of the most unstable mode clearly becomes smaller (larger finger size) when Λ decreases. This trend is consistent with that observed in the other two sediment sizes, but the dependence is stronger for the intermediate size. In the study done by Burns and Meiburg [2012], they did not consider the sediment phase act as fast-diffusing agent in the double-diffusive system; however, they also found that when the settling velocity of sediment particles is large, the Rayleigh-Taylor mode becomes the dominant mode of instabilities.

4.3 Effect of δp on the Linear Stability

[36] The assumption δp = δs is highly idealized. There are many factors that can affect the magnitude of δp and δs, which are beyond the scope of this study. Here, we present one of the probable scenarios in which δp and δs of the base flow are determined by their corresponding diffusion coefficients. We consider the situation where both the sediment and salt concentration profiles evolve from initial Heaviside step functions. After a short period of time, the thickness of the diffusion layer δp (δs) of sediment (salt) concentration will scale with the square root of the sediment (salt) diffusivity. It is reasonable to assume

display math(37)

[37] Since the sediment phase can diffuse slower or faster than salt, δp can be either smaller or greater than δs. The difference in δp and δs in conjunction with higher sediment load can lead to an unstable initial bulk density profile.

[38] For fine sediment with d = 2 µm, the resulting δp calculated based on the scaling law of equation (37) is about five times smaller than that of δs. As shown in Figure 5a, this sharp initial sediment concentration profile greatly enhances the growth rate as compared to that using δp = δs (see Figure 3a). The effect of Stokes settling remains negligible in this scenario due to very small V0. It is important to note that when the sediment-to-salt ratio Λ becomes greater than 0.25 (i.e., higher sediment load with concentration greater than about 10 g/l), the growth rate increases drastically. As shown in Figures 5c and 5d, the initial bulk density profile becomes unstable (bulk density increases upward) just above the interface (z = 0) when Λ becomes greater than 0.25. A statically gravitationally unstable layer will form at the interface; the system becomes similar to the classic Rayleigh-Taylor problem [Voltz et al., 2001]. The instability can occur without the need to conquer the initial negative (stable) density gradient through migration via double diffusion and/or Stokes settling, and the growth rate is much greater. When Λ is sufficiently large to give initially an unstable bulk density profile, the wave number of the dominant mode is greatly reduced. As demonstrated in Figure 5b, when Λ changes from 0.15 to 0.5, the dominant wave number shifts from around 100 cm− 1 to 17 cm− 1. The decrease of the dominant mode indicates a significant increase of the characteristic finger size, consistent with Rayleigh-Taylor instability. For this fine sediment size, there still exists a critical value of the ratio Λcrit, below which the system stays stable.

Figure 5.

(a) Growth rate G as a function of Λ with sediment diameter d = 2 µm; (b) dispersion relationship with Λ = 0.5 and 0.15; (c) bulk density profile with Λ = 0.5; (d) bulk density profile with Λ = 0.15.

[39] For larger grain size with d = 20 µm, the resulting δp by equation (37) is about 6 times greater than that of δs. A direct consequence of δp > > δs is that the initial bulk density profile becomes always unstable for the range Λ = 10− 2 ∼ 1 considered in this study (see Figures 6a and 6c). The overall growth rate is about 3 times greater than that of δp = δs. At Λ = 0.5, the dominant wave number is around 3 cm− 1 (Figure 6b) corresponding to centimeter-scale finger sizes. Due to the presence of the unstable initial bulk density profile, Rayleigh-Taylor instability occurs without the need of sediment particles settling through the density interface [Burns and Meiburg, 2012]. The calculated growth rate by setting Stokes settling velocity V0 to zero is identical to that with Stokes settling included.

Figure 6.

(a) Growth rate G as a function of Λ with sediment diameter d = 20 µm; (b) dispersion relationship with Λ = 0.5; (c) bulk density profile.

[40] Similar features of Rayleigh-Taylor instability are observed for grain size of d = 60 µm (Figure 7), where the resulting δp by equation (37) is about 32 times greater than that of δs and the initial bulk density profile is always unstable. Due to very large Stokes settling velocity in this case, Stokes settling still plays some minor role in the resulting growth rate and dominant wave number. The growth rate with Stokes settling included is slightly smaller than that without Stokes settling. This could be again due to the smearing of the vorticity field [Burns and Meiburg, 2012]. Another important observation is that for large initial sediment load Λ > 0.3, the growth rate becomes smaller compared with that of the scenario of δp = δs. Due to the large diffusion coefficient and the significant increase in δp, the gradient of the initial sediment concentration profile is significantly reduced and the growth rate is reduced. However, at low ratio (Λ < 0.3), the growth rate remains significantly larger than that of δp = δs due to the initial unstable bulk density profile. Compared to Figure 4c, the dominant mode in this case decreases more significantly from 5 cm− 1 to 1 cm− 1, which gives a large characteristic finger size of several centimeters. Again, the decrease of the dominant wave number suggests that the initial presence of unstable layer tends to enhance the Rayleigh-Taylor mode and leads to the formation of relatively larger sediment fingers.

4.4 Effect of ε on the Linear Stability

[41] The present linear stability analysis only allows studying the development of instability at very early stage. During a realistic convective sedimentation, when the growth rate is small, the timescale for the large-scale motion to appear may be sufficiently long such that a noticeable nonzero value of ε is introduced. To fully understand the linear response of convective sedimentation, we are motivated to investigate the effect of ε. The most apparent effect of introducing a finite, positive value of ε is that the initial bulk density profile becomes unstable. This effect is similar to that discussed in section 4.3 with δp ≠ δs and ε = 0, and an enhanced growth rate and larger convective sediment finger size are expected. We specify the base state profiles for non-dimensional sediment and salt concentration to be

display math(38)
display math(39)

with ε(0) = 0, δp(0) = δs(0) = δ.

[42] To put our analysis into a physical context, we consider ε = V0t, where t is the non-dimensional time before the occurrence of the large-scale motions. For flow with a step change in salinity or sediment concentration, the change of the profiles can evolve faster than the growth of perturbations initially. The frozen time analysis is not suited to study the early time behavior of the system [Gresho and Sani, 1971; Ihle and Niño, 2011; Riaz et al., 2006]. In our study, we started our analysis with error function profiles for both the salt and concentration profiles instead of a step change to account the mixing processes in nature. We carried out the quasi-steady analysis to study the effect of the settling. For a quasi-steady analysis to be accurate, the growth of perturbations needs to be faster than the change of the base state. In the present problem, there exist three different time scales, namely the diffusion time scale of salt math formula, the diffusion time scale of sediment math formula, and the settling time scale of sediment math formula. We can estimate these time scales with the given δ value and the properties of the sediment phase as

math image(40)
math image(41)
display math(42)

[43] For δ = 5 mm adopted in this study, math formula for all cases. We can then examine other time scales for different particle size. For d = 2 µm, both the diffusivity of sediment phase and Stokes settling can be calculated as

display math(43)

and this gives math formula and math formula. In this study, we chose a threshold value of growth rate Gt = 0.01 s− 1, and the corresponding ratio Λ is defined as Λt. This gives a time scale for the growth of perturbations to be 100 s. It is evident that math formula, math formula, and math formula are all much greater than 100 s, suggesting that the perturbations grow faster than the change of the base state. On the other hand, for d = 20 µm, both the sediment diffusivity and the settling velocity become large. Following the same analysis, we calculate the corresponding time scales to be math formula and math formula, which are only slightly greater than 100 s. Therefore, for d > 20 µm, the quasi-steady analysis should be applied with caution. However, as we can show later in the detailed discussion, for large sediment particle (d = 20 µm), the growth rate of the fastest growing mode is typically in the order of O(0.1 s− 1), which gives a time scale of 10 s. This means that the quasi-steady analysis is still very accurate even for large sediment particle used in this study. The frozen time scale is a good approximation for all particle sizes considered as far as the fastest growing modes are concerned. The frozen base flow assumption becomes less accurate for larger particles, if we consider near neutral modes.

Figure 7.

(a) Growth rate G as a function of Λ with sediment diameter d = 60 µm; (b) dispersion relationship with Λ = 0.5; (c) bulk density profile.

[44] The quasi-steady base flow is simply the solution of the advection-diffusion equation for sediment phase with a constant Stokes settling velocity and pure diffusion equation for salt concentration with δp(0) = δs(0) = δ = H/20 for the present analysis. And δp and δs are given as

display math(44)
display math(45)

[45] Figure 8a presents the growth rate G as a function of Λ for d = 2 µm with ε = 0, δ/2, and δ.

Figure 8.

(a) Growth rate G as a function of Λ at different vertical distance ε for sediment with sediment diameter d = 2 µm; (b) dispersion relationship with different ε at Λ = 0.5; (c) bulk density profile for different ε.

[46] The corresponding bulk density profiles are shown in Figure 8c where unstable bulk density profiles are observed below z = 0. Compared to the case with ε = 0, the growth rate for ε = δ/2 is about 20 times greater. However, for the case with ε = δ, the growth rate is close to that of ε = δ/2. According to the dispersion relationship shown in Figure 8b, we clearly observe that by introducing a nonzero value of ε, the most unstable mode is significantly reduced from about 100 cm− 1 to 15 cm− 1. This suggests that when ε increases from zero, double-diffusive mechanism becomes less important and the coarsening of sediment fingers is expected. Following Burns and Meiburg [2012], we also examine the vorticity eigenfunctions to provide insights on the change of the instability modes. Figure 9 shows the vorticity eigenfunctions for d = 2 µm with ε = 0 and ε = δ. For ε = 0, dipole vortices can be identified which are nearly symmetric about the interface. However, for ε = δ, with the formation of a gravitational unstable layer, the structure takes on the shape of a quadrupole as the dominant mode shifts from the double-diffusive mode to Rayleigh-Taylor mode. Similar observation is also discussed by Burns and Meiburg [2012].

Figure 9.

Vorticity eigenfunctions for d = 2 µm and Λ = 0.75 with (a) ε = 0 and (b) ε = δ. The contour values are plotted 1 × 10− 3, 1 × 10− 2, and 0.1.

[47] Figure 10 shows the growth rate G as a function of ε for a range of ratio Λ for d = 2 µm. At a given Λ, the growth rate increases rather slowly when ε/δ is small. Because the diffusion process reduces the gradients of both the sediment and the salt concentration profiles, the increase of growth rate for small ε/δ is mild. When ε is sufficiently large, the formation of unstable bulk density profile eventually dominates the growth of instabilities and the increase of the growth rate G becomes more profound with ε/δ. When ε/δ further increases, the increase of growth rate tends to slow down and eventually reaches a plateau. When the density interface induced by the sediment phase is below the density interface (ε = δ) induced by salt, sediments can barely feel the ambient stratification by salt, and the growth rate ceases to increase (in fact starts to decrease) due to the diffusion effect. This trend is consistent with earlier work by Burns and Meiburg [2012].

Figure 10.

Growth rate G as a function of ε with different Λ value for sediment with sediment diameter d = 2 µm.

[48] Results presented in Figure 10 can be further used to qualitatively estimate the minimum sediment-to-salt ratio Λ for the large-scale motion to appear. For instance, if the threshold for growth rate Gt = 0.01 s− 1 is considered to be significant for a given application (i.e., corresponds to a timescale of O(100) s for the instability to occur and ε/δ = 0.072), Figure 10 suggests that when Λ is smaller than 0.2 (corresponds to 7.9 g/L), convective sedimentation is not expected to occur (regardless of ε) for such fine sediment. It should be noted that although Figure 8b indicates a significant increase of characteristic finger size for ε = δ/2 and ε = δ, the critical condition discussed here is of small ε/δ ≈ 0.07 and lower ratio (Λ ∼ 0.2) and model results (not shown) suggest that the dominant wave number is around 70–80 cm−1 and the characteristic finger width remains millimeter scale.

[49] Figure 11a shows the growth rate G as a function of Λ for d = 20 µm with ε = 0, δ/2, and δ. Compared to the growth rate of ε = 0, the growth rate increases significantly for ε = δ/2. As expected, unstable bulk density profile is observed throughout z < 0 (see Figure 11c). The overall growth rate for ε = δ is quite similar to that of ε = δ/2. It is interesting to note that when Λ is greater than 0.1, the growth rate for ε = δ/2 start to becomes slightly greater than that of ε = δ. This is because when ε is sufficiently large, the concentration profile becomes more smooth due to diffusion and the growth rate becomes smaller. More importantly, for the range of Λ considered here, the growth rate G is always greater than about 0.1 for ε = δ/2 and ε = δ, implying that the instability may always occur in practical applications. Because of the negligible effect of double-diffusive mechanism, the dominant mode is around k = 4 cm− 1 regardless of ε (see Figure 11b).

Figure 11.

(a) Growth rate G as a function of Λ at different vertical distance ε for sediment with sediment diameter d = 20 µm; (b) dispersion relationship with different ε at Λ = 0.5; (c) bulk density profile for different ε.

[50] Figure 12 shows the growth rate G as a function of ε for d = 20 µm for a wide range of ratio Λ. At low ratio (Λ < 0.5), the growth rate increases quite rapidly for small ε/δ and then reaches the peak for ε/δ = 0.2 − 1 (depends on Λ) and finally starts to drop slowly. If we consider Gt of 0.01 s− 1 (timescale ∼ 100 s), the resulting ε/δ is around 7. Figure 12 then clearly indicates that convective instabilities will occur even at the lowest ratio Λ = 0.01 (0.4 g/l) studied in this paper.

Figure 12.

Growth rate G as a function of ε with different Λ value for sediment with sediment diameter d = 20 µm.

[51] We have demonstrated a strong dependence of the convective sedimentation on the sediment grain size. To obtain a more clear picture regarding the response of the system on grain size, Figures 13 and 14 present the dependence of growth rate on ε/δ and Λ for grain diameter of d = 4 µm and 10 µm. For grain size of d = 4 µm, the overall growth rate is smaller than that of d = 2 µm. For instance, considering Gt of 0.01 s− 1, the resulting ε/δ is 0.3 and the resulting Λt for the large-scale motion to be observed is around 0.33 (13 g/l), which is greater than that of d = 2 µm (7.9 g/l). It should be note here that for d = 4 µm, the ratio of the diffusion coefficients between the fast-diffusing agent (salt) and the slow-diffusing agent (sediment) is only κs/κ0 = 3.4. The ratio of the two diffusion coefficients is reduced by a factor of 8 compared to that of d = 2 µm; the double-diffusive mechanism is expected to be weaker. On the other hand, the settling velocity is 4 times greater than that of 2 µm; the settling-driven mechanism, therefore, is enhanced. However, the overall behavior of the system is still very similar to the case with d = 2 µm. It can be calculated that at d = 6 µm, the double-diffusive mechanism is almost completely gone. Once the grain size is greater than d = 6 µm, sediment phase becomes the fast diffusion agent and the double-diffusive effect starts to increase with grain size while the settling-driven mechanism also increases. Consequently, for d = 10 µm, the growth rate for the given range of ε/δ and Λ (see Figure 14) becomes significantly larger and is comparable to that of d = 20 µm (see Figure 12).

Figure 13.

Growth rate G as a function of ε with different Λ value for sediment with sediment diameter d = 4 µm.

Figure 14.

Growth rate G as a function of ε with different Λ value for sediment with sediment diameter d = 10 µm.

[52] For all the cases discussed so far, we can summarize that the dynamics of the system is determined by the relative importance of three characteristic length scales: the diffusive thickness of sediment concentration δp, salt concentration δs, and the settling distance of sediment particles ε. Only for very fine sediment of d < 6 µm, with very small settling velocity V0 = O(1), the double-diffusive mode is excited and has enough time to grow at ε << 1. Millimeter-scale sediment fingers are, therefore, expected. For Gt of 0.01 s− 1, the resulting Λt for the large-scale motion to be observed is around 0.2–0.4, which corresponds to sediment load of 8–16 g/l. For coarser sediment d ≥ 10 µm with larger settling velocity, convective instability is expected to occur even at 0.4 g/l, which is the lowest sediment concentration considered in this study; ε quickly increases to O(δ) and Rayleigh-Taylor modes start to dominate with centimeter-scale sediment fingers as expected.

4.5 Scale Analysis

[53] For a double-diffusive system, recent study by Sreenivas et al. [2009] and Burns and Meiburg [2012] showed that the finger width scales with the Rayleigh number to the power of − 1/3 and the growth rate of the most unstable mode scales with the Rayleigh number to the power of 2/3. On the other hand, for pure Rayleigh-Taylor instabilities without diffusion [Chandrasekhar, 1961], the wave number corresponding to the largest growth rate scales with the reduced gravity g′ = gΔρ/ρsalt with ρsalt the saltwater density in the lower layer

display math(46)

and the growth rate of the most unstable mode can be given as

display math(47)

[54] The laboratory observation of Hoyal [1999a] also confirmed the measured finger thickness scales with the reduced gravity g′ with a power law relationship of L ∝ g− 1/3, which also gives k ∝ g1/3. The above relationships can be converted using the dimensionless groups defined in this study, and we obtain

display math(48)

[55] It is noted here that instabilities due to pure double-diffusive or Rayleigh-Taylor mechanism have similar scaling laws. However, as we discussed earlier, the finger size is typically found to be of millimeter scale [Stern, 1960; Hoyal, 1999a; Green, 1987] for double diffusion, and when the settling-driven mechanism (Rayleigh-Taylor instability) dominates [Hoyal, 1999a; Parsons et al., 2001], centimeter-scale fingers are expected. As discussed in the previous sections, when both double-diffusive and settling-driven mechanisms co-exist in the present problem, we expect a transition of observed finger size between the millimeter scale and the centimeter scale as the properties of sediment along with the bulk density profile change. Therefore, we are motivated to investigate the scaling law for the present problem.

[56] Figure 15 shows the maximum growth rate and the corresponding most unstable wave number plotted against the ratio Λ for d = 2 µm. When Λ is large (high initial sediment concentration in the upper layer), the formation of unstable bulk density profile favors the onset of Rayleigh-Taylor instability (see Figure 5c) and the scaling laws for both the most unstable wave number and the corresponding growth rate follow those of pure Rayleigh-Taylor instability. For small Λ, the initial unstable density profile is not strong enough, and with such small grain size, the double-diffusive mechanism becomes dominant with millimeter-scale fingers expected. Indeed, as Λ decreases from a large value, there is a transition of finger size (or the most unstable wave number) from centimeter scale to millimeter scale (see Figure 15b).

Figure 15.

(a) Growth rate and (b) corresponding most unstable wave number for three different ε values with d = 2 µm.

[57] For coarser grain of d = 20 µm (see Figure 16), the maximum growth rate shows a best-fitting slope around 0.57 and the corresponding wave number of the most unstable mode a slope around 0.2. As discussed previously, for d = 20 µm, the settling-driven mechanism dominates and the finger size is always of centimeter scale. Since there is no transition of dominant mechanisms, the resulting scaling laws are closer to the theoretical prediction. Slight derivation from the theoretical predictions is possibly due to the diffusion effects.

Figure 16.

(a) Growth rate and (b) corresponding most unstable wave number for three different ε values with d = 20 µm.

5 Concluding Remarks

[58] Linear stability analysis has been applied to study the convective sedimentation in a stably stratified saltwater. The sediment diameter considered in this paper is in the range of 2 µm to 60 µm with the specific gravity of γ = 2.65, which is in the small Stokes number regime (St < 0.2). This range of grain size is of typical fine terrestrial sediment delivered by rivers in the coastal environment. The sediment concentration studied is in the range of 0.4–39.4 g/l in the ambient flow with the salt concentration of 0.035 (35 ppt) which gives commonly observed hypopycnal flow. Sediment diffusivity κ is determined by long-range interactions between sediment particles and is given by a semi-empirical formula based on experiment results [Segrè et al., 2001]. For the given saltwater with a fixed salt diffusivity, sediment phase can act as either a slow-diffusing agent or a fast-diffusing agent in the double-diffusive system depending on the sediment diameter. It is found that when sediment diameter is very fine (d = 2 µm), the Stokes settling velocity is very small and κs/κ > > O(1) where sediment phase acts as the slow-diffusing agent. The double-diffusive mechanism dominates the instability with millimeter-scale convective finger width expected. Further increase of grain size generally gives weaker double-diffusive effect or greatly enhanced settling-driven effect. The resulting convective instability is similar to the Rayleigh-Taylor instability with centimeter-scale convective sediment fingers. The largest sediment finger width can be of several centimeters, which occurs at high sediment concentration with coarse grain of d = 60 µm. When the grain diameter is greater than around d = 10 µm, convective instabilities can occur at sediment load as low as 0.4 g/l. However, for fine sediment with diameter below d = 10 µm and Gt of 0.01 s− 1, the minimum sediment load for convective sedimentation to be significant is around 10 g/l.

[59] The results and findings reported in this study are based on linear stability analysis which provides insights on the relative importance of double-diffusive and settling-driven mechanisms, the characteristic width of the convective sediment finger and Λt (sediment concentration) for the large-scale motion to be observed. However, linear stability analysis is limited to the study of initial development of the instability. Many practical applications require estimating the amount of sediment deposits [Parsons et al., 2001; Warrick et al., 2008], which further requires information on the sediment flux or the effective settling velocity due to convective sedimentation with a fully nonlinear analysis. To further investigate the fully nonlinear dynamics of convective sedimentation, especially to quantify the enhanced settling rate due to convective instabilities, a 3D numerical simulation is necessary. Such more complicated and comprehensive systems will be the subject of future study.

[60] The effects of ambient shear flow and the associated turbulent mixing are neglected in the present idealized study. In natural river plumes, the existence of Kelvin-Helmholtz instabilities [e.g., Geyer et al., 2010] at the two-layer interface may further affect convective sedimentation. Maxworthy [1999] carried out a series of laboratory experiments on the dynamics of sedimenting surface gravity current with high sediment concentration. They observed centimeter-scale convective sediment fingers and argued that the formation of a shear layer at the bottom of the density current encourages a gravitational unstable layer. Future numerical studies directly incorporating stratified turbulent shear layer in convective sedimentation are needed.

Acknowledgments

[61] This study is supported by National Science Foundation (OCE-0926974; OCE-1130217) and Office of Naval Research (N00014-11-1-0176) to University of Delaware. S. Balachandar is supported by National Science Foundation (OCE-1131016) to University of Florida.

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