Convective instability in sedimentation: Linear stability analysis

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 mm), 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 mm (settling velocity 0.09 mm/s), doublediffusive 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 À1 , the minimum sediment concentration is about 8-15 g/l. For grain size greater than or around 10 mm, 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.

Sedimentation from flocculated suspensions in the presence of settling‐driven gravitational interface instabilities

“…This produces an unstable band of sediment-laden saltwater, i.e., the so-called interface layer, sitting between the muddy freshwater and the underlying clear saltwater. This phenomenon has been experimentally observed by Hoyal et al [1999b], Parsons et al [2001], and Blanchette and Bush [2005] and examined through linear stability analysis by Burns and Meiburg [2012] and Yu et al [2013] and Direct Numerical Simulation (DNS) by Yu et al [2014] and Burns and Meiburg [2015]. A few defining characteristics of the settlingdriven mechanism are the general creation of the interface layer below the initial stratification contact [Hoyal et al, 1999b], the onset of Rayleigh-Taylor generated plume like structures, asymmetry in the downward and upward interface motions, and the overall larger length scales of the instabilities and instabilities spacing (centimeter scale) compared to the double-diffusive generated fingering (millimeter scale) [Hoyal et al, 1999b;Blanchette and Bush, 2005;Burns and Meiburg, 2012;Yu et al, 2013;Burns and Meiburg, 2015;Yu et al, 2014].…”

“…Double-diffusive fingering can also occur in thermally stratified systems with warm suspensions of sediment overlying cooler clear water [Houk and Green, 1973;Green, 1987;Hoyal et al, 1999a;Parsons et al, 2001]. Key in the development of double-diffusive fingering for suspensions of sediment is that the sediment in the buoyant layer settles very slowly so that substance diffusion, and not particle settling, is primarily responsible for changes in the vertical density profiles [Burns and Meiburg, 2012;Yu et al, 2013;Burns and Meiburg, 2015;Yu et al, 2014]. Within the band of conditions conducive to double diffusion, transport of sediment within the doublediffusive fingers can be a significant contribution to the total downward sediment flux [Hoyal et al, 1999a].…”

“…They then extended their analyses with fully nonlinear 2 and 3-D DNS that allow for detailed probing of the interface velocities, concentrations, and fluxes. The initial studies of Burns and Meiburg [2012] and Yu et al [2013] both suggested that when the Stokes diameter of the sediment in the freshwater layer is greater than approximately 10 lm, it is likely that settling-driven convection will dominate at isothermal conditions under salt induced stratification. The 3-D simulations of Yu et al [2014] for three different particles sizes (2, 4, and 20 lm) under a range of concentrations (4-32 g/L) confirmed and further teased out the conditions that will lead to different instability modes by showing that for unflocculated clay (d5 2-4 lm) salt stratification at isothermal conditions should lead to double-diffusive dominated instabilities.…”

“…More recently, the studies of Meiburg [2012, 2015] and Yu et al [2013Yu et al [ , 2014 have been particularly helpful in understanding key differences in initial conditions that will lead to the onset of doublediffusive or settling-driven Rayleigh-Taylor or leaking instabilities in coastal settings. Both research groups used linear stability analysis to examine the first order instability growth rates under a spectrum of initial conditions.…”

“…A few defining characteristics of the settlingdriven mechanism are the general creation of the interface layer below the initial stratification contact [Hoyal et al, 1999b], the onset of Rayleigh-Taylor generated plume like structures, asymmetry in the downward and upward interface motions, and the overall larger length scales of the instabilities and instabilities spacing (centimeter scale) compared to the double-diffusive generated fingering (millimeter scale) [Hoyal et al, 1999b;Blanchette and Bush, 2005;Burns and Meiburg, 2012;Yu et al, 2013;Burns and Meiburg, 2015;Yu et al, 2014]. Further key observations regarding settling-driven instabilities have been that (1) the thickness of the interface layer at the commencement of downward convection tends to thicken as concentration goes down [Hoyal et al, 1999b], (2) that the spacing between points of downward convection tends to also be inversely related to concentration [Hoyal et al, 1999b;Yu et al, 2013], and (3) that the diameters of the instabilities may increase with concentration or sediment settling velocity [Yu et al, 2013].…”