Sediment eddy diffusivity in meandering turbulent jets: Implications for levee formation at river mouths

Authors


Abstract

[1] Depositional patterns characteristic of river mouths are controlled by the sediment-laden turbulent jet exiting from rivers. Here we show that jet instability, developing at river mouths with small width-to-depth ratio, can affect these patterns. Numerical simulations indicate that sediment eddy diffusivity, which is a measure of the spreading of sediments out of the jet core, depends on the interplay between sediment settling and large coherent flow structures, which are associated with unstable jets. Our results suggest that optimal conditions for eddy diffusivity are met when the time scale of sediment settling is equal to the characteristic eddy time scale. In this condition, sediments are deposited at the farthest distance from the jet centerline, thus promoting the formation of lateral deposits. Settling time scales much greater or much smaller than the eddy time scale is associated with low eddy diffusivity, which promotes the formation of a central deposit. Such eddy-mediated sediment transport is more evident in the Zone Of Established Flow than in the Zone Of Flow Establishment, since the former is characterized by fully developed vortices. A simplified model is proposed to estimate the settling and eddy time scales and to predict the optimal condition for levee formation.

1 Introduction

[2] Deltas are often densely populated and are threatened by global change and relative sea level rise [Kim et al., 2009; Ericson et al., 2006]. They host highly diverse ecosystems and provide a natural defense against storms. Moreover, they are among the most productive environments on earth in terms of fossil fuels [Bianchi and Allison, 2009]. It is therefore important to understand the basic mechanisms responsible for delta formation and evolution.

[3] Deltaic deposits are formed by sediment-laden turbulent jets exiting from river mouths [Bates, 1953; Wright, 1977; Wang, 1984; Edmonds and Slingerland, 2007, 2010; Geleynse et al., 2010; Rowland et al., 2009, 2010; Falcini and Jerolmack, 2010; Nardin and Fagherazzi, 2012]. Despite the variability of river mouth depositional features and the large number of external factors that control their formation, two end-member deposits have been recognized: mouth bars and elongated levees [Kim et al., 2009; Edmonds and Slingerland, 2010; Falcini and Jerolmack, 2010; Geleynse et al., 2011]. The former are associated with branching deltaic networks [Wright, 1977; Edmonds and Slingerland, 2007] while the latter are associated with elongated, birdsfoot-like deltas and tie channels [Rowland et al., 2010; Falcini and Jerolmack, 2010]. The pioneering work of Wright [1977] further classified mouth bars in two types: “lunate” bars, which are formed by inertia-dominated effluents, i.e., jets exhibiting low lateral spreading angles, and triangular “middle-ground” bars, which are formed by friction-dominated effluents, i.e., jets exhibiting elevated spreading angles. The study of turbulent jet hydrodynamics and its coupling with sediment transport and deposition is therefore central to the understanding of delta formation and evolution [Rowland et al., 2010; Falcini and Jerolmack, 2010]. In particular, it is crucial to understand under what physical conditions elongated lateral levees form [Falcini et al., 2012].

[4] Plane turbulent jets are inherently unsteady flows. However, for many geomorphological applications, they have been approximated as statistically steady [e.g., Bates, 1953; Wang, 1984; Agrawal and Prasad, 2002; Edmonds and Slingerland, 2010; Falcini and Jerolmack, 2010; Geleynse et al., 2011]. The integral theory of turbulent jets predicts the steady state solution but not its unsteady characteristics [Agrawal and Prasad, 2003]. For example, shear instability at the jet boundaries creates unsteady coherent structures with a horizontal scale on the order of the jet width [Eliassen, 1983; Balmforth and Piccolo, 2001]. These structures are often described as “meanders”, in analogy with the plane configuration of meandering rivers.

[5] Laboratory and numerical experiments showed that meandering structures account for the bulk lateral transport of momentum or, equivalently, for the bulk horizontal eddy viscosity [Nadaoka and Yagi, 1998; Ikeda, 1999; Rogerson et al., 1999; Rowland et al., 2009]. Indeed, horizontal fluid exchange can result from either the detachment of rings and other spin-off eddies or the meandering motions of the stream. Nonetheless, most studies that seek to explain the characteristics of sediment transport and deposition at river mouths assume, explicitly or implicitly, stable jets [Edmonds and Slingerland, 2007, 2010; Geleynse et al., 2010; Falcini and Jerolmack, 2010; Nardin and Fagherazzi, 2012].

[6] A distinct feature of jet hydrodynamics is the presence of two qualitatively different flow regions [Bates, 1953]: the Zone Of Flow Establishment, close to the river mouth (ZOFE), and the Zone Of Established Flow, developing farther in the basin (ZOEF). In the ZOFE, shear layer instability developing at the jet lateral boundaries is the key mechanism for vortex formation [see Thomas and Prakash, 1991 and Thomas and Chu, 1992], while in the ZOEF, the interaction between the vortices formed on the lateral shear layers becomes dominant. Whether these different hydrodynamic zones have a consequence on the depositional patterns is not yet fully understood.

[7] Novel insights regarding the dynamics of meandering turbulent jets and their associated depositional patterns come from Rowland [2007], who found that a very high sediment eddy diffusivity in a meandering jet was needed to explain the formation of lateral levees. The author hypothesized that this high diffusivity, namely the significant difference between lateral diffusivity of momentum and lateral dispersivity of sediment, was a consequence of the interaction between the meandering structure and the vertical sediment dynamics (i.e., sediment settling). When the depositional time scale is much larger than the eddy time scale, sediments are transported inward and outward by the meandering structures, leading to a small eddy diffusivity. When the depositional time scale is equal or smaller than the eddy time scale, sediments predominantly experience the outward phase of the eddy, leading to a high eddy diffusivity [Rowland, 2007; Rowland et al., 2010].

[8] Inspired by these observations, we further explore this hypothesis by using a numerical hydromorphodynamic model. The goals of this work are (1) to determine the basic mechanism that drives sediment eddy diffusivity in a meandering turbulent jet, (2) to identify and quantify the relevant scales that control the process, and (3) to determine the consequences for sediment deposition patterns.

2 Numerical Modeling of a Sediment-Laden Plane Turbulent Jet

2.1 Shallow Water Equations

[9] A plane turbulent jet flowing into a large basin with still water and a flat bottom is modeled by the unsteady shallow water equations, implemented in the Delft3D software package [Lesser et al., 2004]. The study is restricted to conditions without vertical stratification, caused by either salinity or suspended sediments. As such, the velocity profile is well approximated by a logarithm, and it is parametrized with a depth-averaged value. The two-dimensional schematization of the flow is in accordance with the observation that river effluents are well represented by a planar jet [Van Prooijen and Uijttewaal, 2002; Rowland et al., 2010; Landel et al., 2012]. Such schematization is also supported by studies that used a depth-averaged formulation to reproduce large coherent structures in nonstratified river flows [Nadaoka and Yagi, 1998; Ikeda, 1999].

[10] Neglecting density and atmospheric pressure gradients as well as volume forces resulting from Earth's rotation and other external stresses associated with wind-generated waves, the horizontal momentum and continuity equations read

display math(1)
display math(2)

where ux and uy are the velocity components in the x and y directions, respectively, t is time, η is the water level with respect to a reference plane, d is the water depth, g is the gravitational acceleration, Cz is the Chézy's friction coefficient, and inline image is the horizontal turbulent eddy viscosity. The Chézy coefficient is assumed space and time invariant (Cz = 55 m1/2s−1, corresponding to a drag coefficient cd≈ 0.003). The eddy viscosity is computed through a Horizontal Large Eddy Simulation (HLES) which accounts for the damping of subgrid eddies by bed friction, and it is a function of the strain rate of the fluctuating velocities [Uittenbogaard, 1998; Van Vossen, 2000] (see supporting information).

[11] This formulation does not simulate the effect of the meandering coherent structures, which are a peculiar characteristic of unstable jets and occur at spatial scales on the order of the jet width [Van Prooijen and Uijttewaal, 2002]. An example of the distinction between the two types of turbulence is described by Nadaoka and Yagi [1998] and Ikeda [1999].

2.2 Sediment Transport Equations

[12] The transport of various grain size classes is quantified by means of Van Rijn's formulations for bed load [Van Rijn, 1984a] and suspended load [Van Rijn, 1984b] of noncohesive sediment. The bed load transport formulation is given by [Van Rijn, 1984a]

display math(3)

where qb is the bed load sediment flux, ss=ρs/ρw is the relative density, with ρs and ρw being the grain density (here quartz: 2650 kg m−3) and water density (here: 1000 kg m−3), respectively, d50 is the median grain diameter, and D50 is a dimensionless particle parameter [Ackers and White, 1973; Yalin, 1977; Van Rijn, 1984a]

display math(4)

where υ is the kinematic viscosity of water. T is a dimensionless transport stage parameter [Ackers and White, 1973; Yalin, 1977; Van Rijn, 1984a]

display math(5)

where τb is the bed shear stress associated with the grain friction

display math(6)

with inline image, a is a reference level or (equivalent Nikuradse) roughness height, often considered as the thickness of the bed load transport layer (here a = 0.1 m), and τb,cr is the critical bed shear stress derived from the van Rijns approximation of the Shields curve.

[13] Suspended load is computed through the depth-averaged advection-diffusion equation [Van Rijn, 1984b]

display math(7)

where c is the suspended sediment concentration, inline image is the sediment eddy diffusivity, set equal to inline image, where Pr = 0.7. Ts is an adaptation time that, according to Galappatti and Vreugdenhil [1985], is on the order of d/ws, where ws is the sediment settling velocity and ceq is the local equilibrium depth-averaged suspended sediment concentration

display math(8)

where ca is the suspended sediment concentration at height a above the bed

display math(9)

and fs is a correction factor based on the Rouse formulation [Van Rijn, 1984b]

display math(10)

where Z is a modified Rouse number [Coleman, 1970].

[14] The use of the fluid velocity ux and uy in the advection-diffusion equation indicates that sediments are transported solidly with the fluid, i.e. their inertia is negligible.

[15] As expressed by the well-known sediment mass continuity or Exner equation [e.g., Parker, 2004], the imbalance of actual and equilibrium suspended sediment concentration and the spatial gradients in bed load transport rate govern changes in bed elevations. In the simulations where morphology is updated, a multiplicative morphological scale factor, ranging from 1 to 10 depending on the hydrodynamic conditions, is used to reduce the simulation time. The main parameters used in the model are summarized in Table 1.

Table 1. Constant Parameters Used in the Simulations
ParameterMeaningValue
PrRatio Between A Priori Eddy Viscosity and Diffusivity0.7
ρWater Density1000 kg m−3
CChézy Coefficient55 m1/2s−1
aReference Level for Sediment Transport0.1 m
νWater Kinematic Viscosity10−6 m2 s−1

2.3 Numerical Schematization, Boundary Conditions, and Parameters Setting

[16] Two types of simulations are performed: “frozen-morphology”, in which the bed morphology is kept constant, and “updated-morphology”, in which the morphodynamic feedback is accounted for. The former is used to investigate the detailed sediment transport in a meandering turbulent jet, while the latter sheds light on the resulting depositional patterns.

[17] The shallow water equations are discretized on a staggered finite difference grid and solved by means of an alternate direction implicit time integration method [Stelling and Leendertse, 1992]. The spatial domain is rectangular with dimensions 230 by 340 m; it is spanned by a cartesian grid with cell sizes ranging from 1 m near the outlet of the jet to 10 m at the domain boundaries (Figure 1). The x axis is placed along the jet axial direction with origin at the outlet, while the y axis is transverse to the jet, originating on the jet centerline. A uniform flow with velocity Um is imposed at the jet outlet of width B. A no-flux condition is set on the remaining portion of the outlet boundary, and the water level is fixed at the other three boundaries. The bed elevation next to these boundaries (a strip of 50 m) is set 10 m below the rest of the domain (Figure 1). These dissipation pools are analogous to the experimental setting used by Rowland et al. [2009, 2010] and are used for numerical stability. Finally, a no-reflection condition is imposed on the lateral boundaries, to damp possible seiches.

Figure 1.

Computational domain, initial bed elevation, and boundary conditions. The equilibrium concentration is imposed at the outlet, such that erosion and deposition are equal. The domain size is 340 m in the x direction and 230 m in the y direction.

[18] In view of stability and accuracy constraints of the numerical scheme implemented in Delft3D [Lesser et al., 2004], the computational time step is set equal to 0.6 s. Frozen-morphology simulations are run for 11 h, and outputs are stored every 3 s for analysis. Updated-morphology simulations are run until a central bar develops (see section 2.4.2), and outputs are stored every 5 min. In both types of simulation, the first hour is discarded to account for spin-up effects.

[19] Different combinations of outlet width (B), outlet flow depth (D), and inflow velocity (Um) are used in the simulations (Table 2). Accordingly, the Froude number ranges from 0.16 to 0.64, while the channel aspect ratio ranges from 5 to 10.

Table 2. Geometric and Hydrodynamic Parameters used in the Simulations a
SimulationUm [m/s]D [m]B [m]FrB/W
  1. aAll simulations are performed with five classes of sediments with d50 equal to 64, 96, 144, 216, 324 μm with ρs=2650 kg/m3, and with a passive tracer.
10.5160.166
21.0160.326
31.5160.486
42.0160.646
52.0260.453
61.02100.235
72.01100.6410
82.02100.455

[20] Five classes of noncohesive sediments are considered; d50 = [64, 96, 144, 216, 334] μm with a normalized density ss equal to 2.65, corresponding to a settling velocity of [3.5, 8.0, 14.6, 27.9, 46.5] mm/s, respectively [Van Rijn, 1984b]. In addition, a passive tracer is introduced, which can be considered as a sediment class with zero settling velocity. At the outlet, an equilibrium suspended sediment concentration (equation (8)) is imposed for each sediment class, such that erosion and deposition compensate each other. Sediment density effects are not considered; hence, the hydrodynamics is completely decoupled from the sediment dynamics.

[21] Dimensionless numbers, temporal and spatial scales, as well as sediment characteristics presented herein, are consistent with natural systems that are morphologically associated with turbulent jets outflows, such as tie channels and elongated deltas [Rowland et al., 2009, 2010; Falcini and Jerolmack, 2010] (Table 2).

2.4 Analysis of the Results

2.4.1 Frozen-Morphology Simulations

[22] The frozen-morphology simulations are used to study the detailed hydrodynamics and sediment transport in turbulent jets. For each hydrodynamic and sediment condition, the suspended transport is found to be approximately two orders of magnitude larger than bed load. Therefore, the analysis is only focused on the former transport mode. The velocity and suspended sediment concentration fields are averaged over the entire simulation time, obtaining U(x,y) and C(x,y). The fluctuating flow velocities and suspended sediment concentrations are then computed as u=uU and c=cC.

[23] The power spectrum of uy′ on the principal shear layer y=B/2 (i.e., uy′(x,y=B/2)) is computed using a Hamming window and subdividing the time series in 100 subintervals. Because the results are sampled every 3 s, the spectrum does not give any information about the turbulent energy cascade. In fact, while the time scale of the coherent structures, ToB/Um, is on the order of 10 s, the Kolmogorov scale, ∼To(UmB/ν)−1/2, is on the order of 10−3 s, where ν is the water kinematic viscosity. However, the spectral analysis is still useful for recognizing the characteristic frequency of coherent structures. For this purpose, the peak frequency, fp, is computed using the fifth moment of the spectrum [Young, 1995]. The normalized frequency, fpB/Um, corresponds to the Strouhal number, St. Even though St is usually associated with vortex shedding, it can be used to define the frequency of generation of vortical structures by other, similar mechanisms, e.g., by shear layer instabilities [Thomas and Prakash, 1991] and Kelvin-Helmholtz instability [Van Prooijen and Uijttewaal, 2002].

[24] The y component of the fluid momentum eddy diffusivity (i.e., eddy viscosity) is computed as

display math(11)

For each sediment class, the y component of the sediment eddy diffusivity is computed as

display math(12)

In order to reduce numerical noise, we set a limit on the value of the denominators in equations (11) and (12). We discard values smaller than 10−6 times the maximum value reached by the denominator in the entire spatial domain.

[25] It is important to emphasize that there are two types of eddy viscosity and diffusivity considered in this study. The a priori eddy viscosity and diffusivity, inline image and inline image, are calculated during the simulations using the HLES (see supporting information) and used in equations (1) and (7). These values are used to account, by mean of a diffusion mechanism, for the advection processes which are not explicitly resolved. The a posteriori eddy viscosity and diffusivity, νe and ks, result from the time averaging of the unsteady flow component and are computed at the end of each simulation from equations (10) and (11). The a posteriori viscosities are parameters that synthetically describe the effects of the meandering structures (that are resolved in the simulations) on the lateral transport of momentum and suspended sediments. The a posteriori eddy viscosity and diffusivity are such that, when combined with the time-averaged steady flow, would produce the same momentum and suspended sediment spatial distribution of the unsteady flow. Hereafter, we will refer to the a posteriori eddy viscosity and diffusivity unless differently stated.

2.4.2 Updated-Morphology Simulations

[26] Following Edmonds and Slingerland [2007], we perform morphological simulations until a central deposit forms, eventually leading to channel bifurcation. By using the Edmonds and Slingerland [2007] criterion, the incipient bifurcation is defined when the water depth at any point along the centerline becomes smaller or equal 0.4 D. At this condition, the flow reaches stagnation, such that the flow around the central deposit is greater than the flow above it. The distance of the highest point on the central deposit from the channel mouth defines the parameter Lrmb.

[27] We then introduce an adimensional parameter that quantitatively describes the deposit morphology,

display math(13)

where h(x,y) is the height of the deposit with respect to the initial bed level. This parameter indicates the amount of sediment stored in the levees per unit of levee length compared to the amount of sediment deposited in the channel per unit of channel width. Hence, this parameter indicates the jet capability to transport sediments away from the centerline. High values of MLB indicate an elongated depositional shape with a small central deposit while low values are associated with a large frontal deposition.

3 Results

3.1 Hydrodynamic Results

3.1.1 Jet Instability

[28] A criterion for the instability of turbulent plane jets based on the channel width-to-depth ratio reads [Van Prooijen and Uijttewaal, 2002; Socolofsky and Jirka, 2004; Rowland et al., 2009]

display math(14)

Using the reference value for cd (=0.003), the jet is unstable if the width-to-depth ratio is smaller than 80. Since the maximum width-to-depth ratio used in the Delft3D simulations is 10 (Table 2), the simulated flow field is unsteady and displays large-scale meanders (Figures 2a and 2b). The long-term averaged flow field (Figures 2c and 2d) resembles the steady flow properties that are well described by the integral jet theory [Agrawal and Prasad, 2003].

Figure 2.

(a, b) Snapshot of a simulation. (a) Velocity field, plotted on a coarse grid for graphical purpose. (b) Sediment concentration field normalized by the equilibrium concentration at the outlet, Ce. The white box indicates the region reproduced in Figure 10 and discussed in section 4.2. (c, d, e) Time-averaged (10 h) flow and concentration fields. (c) Ux/Um, (d) Uy/Um, and (e) C/Ce. Simulation parameters: B=6 m, D=1 m, Um=2 m/s, d50=64 μm.

[29] The magnitude and spatial patterns of the root-mean-square of the fluctuating x and y velocities (Figures 3a and 3b) are similar to those measured in laboratory experiments [cf. Rowland et al., 2009, Figure 11], suggesting that the spatial and temporal discretization is adequate to capture the unsteady components of the flow. It can be noted that the flow is not self-similar, at least up to distances considered to be important for the depositional patterns, e.g., x/B<10 [Falcini and Jerolmack, 2010].

Figure 3.

Normalized root-mean-square fluctuating velocities: (a) rms(ux′), (b) rms(uy′). For comparison, see Figure 11 in Rowland et al. [2009]. (c) A posteriori normalized eddy viscosity. For comparison, see Figure 14 in Rowland et al. [2009]. The dashed line is the value provided by the jet integral solution of Agrawal and Prasad [2003], equal to 0.023. Simulation parameters: B=6 m, D=1 m, Um=0.5, 1, 1.5, 2 m/s. Results are averaged using the axis of symmetry y=0, so that the output is reported only for y>0.

3.1.2 ZOFE and ZOEF

[30] In order to investigate the difference between ZOFE and ZOEF, we consider two indicators: the decay of Ux along the centerline, inline image, and the Strouhal number. For simplicity, we assume that the ZOFE includes the transitional zone recognized by Rowland et al. [2009].

[31] Even though inline image monotonically decreases from the channel mouth, two different zones can be individuated: one in which the decay rate is relatively slow and the along-stream profile of inline image is convex, and one in which the decay rate is fast and the along-stream profile of inline image is concave (Figure 4a). The transition between the two zones is at x/B∼4−8.

Figure 4.

(a) Along-jet velocity at the centerline, Ux(x,y=0). Symbols for different simulations are not reported for graphical clarity. (b) Normalized peak frequency (Strouhal number) of the uy′(x,y=B/2) spectrum as a function of the along-jet distance. The dashed line is the minimum Strouhal number predicted by Dracos et al. [1992], St,o=0.07.

[32] A similar transition can be individuated by the Strouhal number, which decreases with increasing distance from the outlet (Figure 4b). A remarkable decrease in St occurs at x/B∼4−8. After this transition, St attains a value close to St,o=0.07, which coincides with theoretical [Dracos et al., 1992] and experimental [Landel et al., 2012] results for the minimum Strouhal number in plane turbulent jets. The higher values of St in the ZOFE are associated with vortices that form at the jet interface (for yB/2) as a result of shear layer and Kelvin-Helmholtz instabilities. These structures scale with the channel half width B/2 as indicated in Figure 2. The lower values of St in the ZOEF indicate structures propagating with a slower advection velocity and/or larger length scale. Indeed, in Figures 2a and 2b, we observe a single coherent motion in the ZOEF, with vortexes characterized by a length scale of ∼B. These large vortexes result from the interaction and merging of the structures formed along the two shear layers delimiting the jet. This interaction has been described in terms of vortex mutual induction [Crow, 1970; Saffman, 1992] and vortex pairing [Winant and Browand, 1974; Saffman and Baker, 1979].

[33] The classification into ZOFE and ZOEF regions can be thus interpreted in terms of large-scale vortex dynamics, i.e., eddy formation and eddy interaction/merging. Unfortunately, this classification is not precise because, as suggested by Browne et al. [1979], the transition between the eddy-forming and eddy-merging region is not sharp.

[34] Based on these observations, we set the transition between the two zones at x/B=6, in accordance with previous definitions [Bates, 1953; Tennekes and Lumley, 1972]. We operatively define the ZOFE as 0<x/B<6, and the ZOEF as 6<x/B<12, and we restrict the analysis to the region |y|/B<2. Varying the transition zone between 4 to 8 B does have a quantitative influence on the results but does not change the qualitative picture presented herein.

3.1.3 Eddy Viscosity

[35] The magnitude of the normalized eddy viscosity νe,y/(BUm) agrees with experimental results [cf. Rowland et al., 2009, Figure 14], and with the theoretical prediction for its maximum value provided by the jet integral solution of Agrawal and Prasad [2003], i.e., ∼ 0.023 (Figure 3c). This suggests that the Delft3D simulations reproduce reasonably well the meandering coherent structures and their effect on the turbulent diffusion of momentum. Noticeably, the normalized eddy viscosity does not depend on velocity (only a minor increase with Um is detected), implying that Reynolds similarity holds.

[36] It is important to note that this eddy viscosity is not a consequence of the HLES. In fact, the normalized a priori eddy viscosity computed with the HLES is relatively small (<∼0.001). We underline again that the a posteriori eddy viscosity results by time averaging the large coherent structures, which are not taken into account with the HLES.

[37] The eddy viscosity decreases along the y direction moving away from the jet centerline and increases along the x direction moving away from the outlet (Figure 3c). The lateral decrease is associated with the damping of shear layer instability away from the jet region. The increase in the streamwise direction is related to the persistence of the large-scale meandering structures that are able to sustain themselves even if the mean momentum and the mean shear are diminishing (consistent with the ZOEF classification). The role of large-scale coherent vortices in the momentum flux is emphasized by the larger average value of the eddy viscosity in the ZOEF (νe,y∼0.01–0.02), as compared with the ZOFE (∼0.005) (Figure 5a).

Figure 5.

Normalized eddy (a) viscosity and (b) diffusivity. Simulation parameters: B=6 m, D=1 m, Um=2 m/s, d50=64 μm.

3.1.4 Sediment Eddy Diffusivity

[38] The spatial pattern of sediment eddy diffusivity (equation (12)) is similar to that of the eddy viscosity: it decreases in the y direction, far from the centerline, and increases in the x direction, away from the jet outlet (Figures 5 and 6). As for the eddy viscosity, the eddy diffusivity is larger in the ZOEF than in the ZOFE (Figures 5 and 6).

Figure 6.

Normalized sediment eddy diffusivity for various sediment sizes, spatially averaged in the ZOFE and ZOEF. B=6 m, D=1 m. (Top) Um=0.5 m/s; (bottom) Um=2 m/s. Results are averaged using the axes of symmetry y=0, so that the output is reported only for the domain y>0.

[39] As a first attempt to relate the hydrodynamics of unstable jets to their depositional patterns, we focus the rest of our analysis on the spatial-averaged values in the ZOFE and in the ZOEF. This simplified approach allows us to synthetically explore a wide range of hydrodynamic and sedimentary conditions.

[40] The sediment eddy diffusivity is much greater than the eddy viscosity (Figure 5), suggesting that sediment concentration disperses more than momentum due to the effect of the meandering structure and its associated lateral eddies (Figure 2). The physical mechanism responsible for this elevated sediment eddy diffusivity involves particle dynamics in the meandering jet and in the large-scale vortical structure: particle settling plays the mayor role [Rowland, 2007], while mechanisms of particle turbulent interaction (e.g., crossing trajectory effects discussed by Elghobashi [1993] and Guala et al. [2008]) induce second order effects.

[41] The important role played by particle settling is clearly shown by the normalized sediment eddy diffusivity, which depends on both sediment diameter and jet velocity. For low velocity (Um=0.5 m/s), the maximum (spatially averaged) eddy diffusivity is reached with fine sediments (d50=64 μm) and decreases with increasing sediment size; for higher velocity (Um=2 m/s), the eddy diffusivity initially increases with grain size, reaching a maximum at d50=144 μm and then decreases with increasing grain size (Figure 6).

[42] To explore this mechanism, we compare the space-averaged eddy diffusivity (in both the ZOFE and ZOEF) varying both velocity and sediment diameter (Figure 7). The grain size associated with the maximum eddy diffusivity increases with jet velocity, especially in the ZOEF. This suggests that, in order to quantify sediment eddy diffusivity, particle settling must be coupled with particle transport, through mean advection and particle trapping in meandering coherent structures.

Figure 7.

Normalized sediment eddy diffusivity, spatially averaged in the ZOFE and in the ZOEF, as function of sediment diameter and jet velocity. B=6 m, D=1 m.

3.2 Morphological Results

[43] Given a fixed set of hydrodynamic conditions, different depositional patterns are associated with different grain sizes (Figure 8). The deposit associated with slow-settling sediments (d50=64 μm) is spread over a wide fanlike area in the ZOEF; lateral deposits are present even though they are not concentrated along the channel sides (Figure 8b). Contrastingly, the bottom morphology associated with fast-settling sediments (d50=324 μm) is characterized by both thin levees and a shallow central deposit in the ZOEF (Figure 8f). Sediments of intermediate grain size (d50=96 μm) result in thick and elongated levees, with a relatively small central deposit in the ZOEF(Figures 8c and 8d).

Figure 8.

Height of the deposit normalized by the outlet depth D at the stagnation point. Simulation parameters: B=6 m, D=1 m, Um=2 m/s. (a) d50=64 μm and ss=1.5, (b) d50=64, (c) d50=96 μm, (d) d50=144 μm, (e) d50=216 μm, (f) d50=324 μm, (g) d50=1094 μm.

[44] To further clarify this trend, we perform two additional simulations: one with a very slow-settling sediment (d50=64 μm, ss=1.5, ws=1.1 mm/s) and one with a very fast-settling sediment (d50=1092 μm, ws=123.5 mm/s). The former condition leads to deposits spread over a large area (Figure 8a), while the latter leads to a mouth bar deposit with almost no levees (Figure 8g). Although all the presented morphologies are obtained from very different sediment grain sizes, the resulting deposits can be identified as “lunate” [Wright, 1977], where both central and lateral deposition occurs simultaneously at different rates.

[45] For the specific set of hydrodynamic conditions considered in Figure 8, the maximum value of MLB, i.e., the ratio between the volumes of the lateral and central deposits, is attained with d50=96 μm (MLB=0.38) while lower values are obtained with either heavier (MLB=0.15) or lighter (MLB=0.25) sediments.

[46] In all runs, levees start to form in the ZOFE (Figures 9a, 9b, and 9c), in accordance with experimental and theoretical results [Rowland et al., 2010; Falcini and Jerolmack, 2010]. However, large levees also form in the ZOEF, which is characterized by large meandering structures even after the bottom has been modified by the flow (Figures 9d, 9e, and 9f). The ZOEF is also the area where, for different hydrodynamic and geometric conditions, a great diversity of central deposits is found, while the ZOFE always remains devoid of any central deposit.

Figure 9.

Evolution of bottom morphology and sediment concentration in time. (a, b, c) Height of the deposit normalized by the outlet depth D. (d, e, f) Sediment concentration field normalized by the equilibrium concentration at the outlet. Simulation parameters: B=6 m, D=1 m, Um=2 m/s, d50=96 μm.

4 Discussion

4.1 Role of Meandering Coherent Structures on Sediment Eddy Diffusivity

[47] The a posteriori sediment eddy diffusivity coefficient takes into account all the processes that are not represented in the steady flow. Our results show that the sediment eddy diffusivity is up to 10 times the eddy viscosity in both ZOFE and ZOEF, and it depends on sediment settling velocity.

[48] Rowland [2007] explained this high eddy diffusivity by recognizing that, in contrast to other tracers (i.e., salinity, temperature, dissolved biochemical compounds), sediment has a vertical dynamic. As a result, in addition to horizontal mixing, sediment can settle and exit the water column.

[49] Sediment eddy diffusivity in a meandering jet can be elucidated by analyzing the trajectory of a sediment particle while it is advected in a meander [Rowland, 2007]. Eddies move the particle from the jet centerline region toward the lateral region and then back to the centerline. If sediments were a tracer without vertical dynamics, such as temperature or salinity, variations in their concentration would be determined by horizontal diffusion, i.e., turbulent mixing, only. Instead, particles are able to settle, especially in the lateral region, where the bed shear stress (or turbulence) is lower. If a particle settles, it will not come back with the inward eddy. Macroscopically, after integration over time and space, this results in a sink of sediments that yields an elevated eddy diffusivity.

4.2 A Conceptual Model for Sediment Eddy Diffusivity

[50] Assuming that the coupling between eddy-induced sediment advection and particle settling is the leading mechanism for the increase in eddy diffusivity, we propose a simplified model for estimating the maximum eddy diffusivity in a sediment-laden meandering jet. Although we are aware that sediments spread out radially from an eddy because of their own inertia, our approach assumes that sediments are horizontally advected, solidly, with the eddy. This is also the assumption adopted by the Delft3D model.

[51] The proposed model is based on the comparison between two time scales: the deposition time scale, TD, and the eddy time scale, TE. The former is the average time needed for a particle to settle, the latter is the average time needed for an eddy to make a half revolution. Because sediments are transported by a range of eddies, there will be a range of eddy time scales as well. Our simplified approach assumes that the sediment dynamic can be described by a single eddy, characteristic of the flow region considered.

[52] This model is inspired by some insights presented in Rowland [2007], who found that high eddy sediment diffusivity was obtained when the settling time scale was comparable to the eddy time scale. In this scenario, “sediment only experienced the outward directed cross-stream velocity and thus did not experience the averaging effect of reversing cross-stream flow directions.” On the other hand, “if settling time is much greater than the turbulence timescale, an individual sediment particle moves back and forth with the turbulence many times before reaching the bed.” Using this scheme, Rowland [2007] concluded that “for sediment with settling time scales significantly greater than the meandering time scale this effect [the increase in eddy diffusivity] should decrease.” The conceptual model of Rowland [2007] can be hence summarized as follows: eddy diffusivity is minimum (i.e., equal to the eddy viscosity) for TD >> TE and increases with decreasing value of TD/TE.

[53] Here we expand and formalize the insights of Rowland [2007] by considering three reference cases. If TDTE, sediments settle exactly after the eddy has made half revolution and hence, the particles are deposited at the farthest distance from the jet centerline (Figure 10b). This is the optimum case that results in the maximum sediment diffusivity.

Figure 10.

Conceptual model of sediments transported on an eddy. (a) For TD<TE, sediments are deposited as soon as they exit the centerline. (b) For TDTE, sediments are deposited at the extreme end of the eddy. (c) For TD>TE, particles are deposited after the eddy has begun the inward phase or are not deposited at all. The figures on the bottom are snapshots of the concentration field (d50=324, 144, 64, μm) at the same time, with D=1 m, Um=2.0 m/s, B=6 m (see Figure 2). Color scale of the concentration is normalized by the equilibrium concentration at the outlet.

[54] If TD>TE, sediments are not fully deposited during an eddy's half revolution and are returned back by the inward component of the eddy. The amount of sediment advected outward is nearly equal to the amount of sediment advected inward. This scenario is defined as “mixing-limited” (Figure 10c).

[55] If TD<TE, sediments are deposited before the eddy has covered half revolution and hence, they do not fully exploit the eddy advection component. The amount of sediment advected in the outward direction is smaller than for the optimum case. This scenario is defined as “mobility-limited” (Figure 10a).

[56] The mixing-limited scenario coincides with the case recognized by Rowland [2007], in which sediments are not captured by a single eddy but are moved back and forth by many eddies. On the other hand, the mobility-limited condition is a novel condition proposed in this work and, therefore, constitutes an extension of Rowland [2007] framework.

[57] For all the conditions explored, the ZOFE remains devoid of any central deposit (Figure 8), suggesting that in this region, the transport capacity is in equilibrium with the sediment flux. Hence, the model should be applied to the ZOEF only, where central deposition can occur.

4.3 Scaling for the Eddy Diffusivity

[58] The ZOEF is characterized by fully developed eddies (Figures 9d, 9e, and 9f), as showed by the relatively constant Strouhal number (Figure 4). Such observation is used to simplify the quantification of the temporal scale TD and TE in this region. We evaluate the deposition time scale as

display math(15)

where LD is the reference elevation for the suspended sediment. For a uniform distribution of sediment within the water column, the reference elevation is equal to D/2. If a Rouse profile is considered, the reference elevation is be located closer to the bed. The Rouse profile reads [Rouse, 1937]

display math(16)

where c is the suspended sediment concentration, ca is the reference concentration at a height z=a of the saltation layer, inline image is the Rouse number with k the von Karman constant (=0.4), and u the bed shear velocity. The resulting reference depth elevation reads

display math(17)

[59] In theory, the shear velocity and the water depth vary in space and time. For simplicity, we use a single Rouse number, obtained by considering the water depth D and the shear velocity computed with the velocity Um.

[60] The eddy time scale can be computed using the frequency analysis of the coherent structures. The characteristic time scale for the eddies in the ZOEF is estimated by considering a reference value for the Strouhal number, equal to 0.07 [Van Prooijen and Uijttewaal, 2002],

display math(18)

[61] Because of the spatial variability in the ZOEF and the presence of a range of eddy scales, TE should be considered as a characteristic scale rather than the exact value for all eddies. As a result, all the predictions based on the time scale ratio are subject to large uncertainty and we do not expect to perfectly match the prediction of the simplified model.

[62] The normalized eddy diffusivity has a clear relationship with the ratio TD/TE (Figure 11a): all the curves related to a fixed set of hydrodynamic conditions (i.e. a set of parameters B, D, Um), have an absolute maximum, equal to ∼0.07, in correspondence of TD/TE ∼1. In addition, all curves seem to collapse on a single envelope, monotonically decreasing for both TD/TE >1 and for TD/TE <1.

Figure 11.

(a) Normalized a posteriori eddy diffusivity (see equation (12)), spatially averaged in the ZOEF, versus the ratio between the sedimentation scale (TD) and the eddy scale in the ZOEF (TE) (see equations (15) and (18)). A maximum is found in the optimum region, defined as 0.5<TD/TE<2. The normalized a posteriori eddy viscosity is plotted as horizontal lines for the various simulations since it does not depend on the sediment type. (b) Ratio between the volumes of the lateral and central deposits, MLB, versus TD/TE. A maximum is found in the optimum region. Simulation parameters as described in Table 2. The labels next to the circles refer to the simulations in Figure 8. The filled dots refer to the only two simulations performed with d50=64 μm and ss=1.5, as in Figure 8a, and d50=1094 μm, as in Figure 8g.

[63] For TD/TE<1, the normalized eddy diffusivity is smaller than the maximum values and approaches zero for TD/TE=0 (Figure 11a). Indeed, for very small values of TD/TE, sediments settle as soon as they exit the centerline and are not transported laterally by the eddy. This result is not surprising, since it is intuitive that heavier sediments, relative to the flow transport capacity, are more difficult to be moved far from the source.

[64] For TD/TE> 1 (Figure 11a), the normalized eddy diffusivity decreases, but it seems to reach a plateau, which is higher than the normalized eddy viscosity. For a passive tracer, characterized by TD/TE=, the ratio between the a posteriori eddy diffusivity and viscosity is about 4. Because a passive tracer lacks a settling behavior, its eddy diffusivity is determined by the eddy dynamic only. The difference between the eddy viscosity and diffusivity is due to the different dynamic of passive tracers and momentum in a turbulent jet: the former can only be advected and diffused, while the latter can be locally reduced by bed friction. While the passive tracer is constantly transported away from the jet, momentum will eventually “sink”. As a result, when computed a posteriori, the diffusivity of the passive tracer is greater than the diffusivity of momentum. Other studies of turbulent jets reported a ratio between eddy diffusivity and viscosity ranging from 1.2 to 2 [Abramovich, 1963; Singamsetti, 1966; Schlichting, 1968]. The higher ratio found in our experiment is likely due to the presence of the large-scale meanders.

[65] As an additional supporting argument for the conceptual model, we propose a simplified explanation for the maximum value of eddy diffusivity in the ZOEF, corresponding to the case with TD=TE. The time-averaged concentration on the jet centerline, Cc(x), decreases with the distance from the outlet. During the outward phase of the vortex, the concentration Cc(x) is moved outward with velocity rms(uy′). During the inward phase of the eddy, a concentration ∼ 0 is moved inward, because we assumed that in the optimum case, all sediments settle at the end of the half revolution. Therefore, the average of the fluctuating term is inline image. The value of rms(uy′)/Um ranges from 0.1 to 0.2 (Figure 3), and a reference value of 0.15 is here considered. The lateral gradient in sediment concentration is dC/dyCc(x)/B. Therefore, the maximum eddy diffusivity is estimated as

display math(19)

which matches the results from the simulations (Figure 11b). This value is about 5–10 times higher than the eddy viscosity (∼ 0.01 BUm) and agrees with the results of Rowland et al. [2009], who predicted an eddy diffusivity 5–20 times higher than the eddy viscosity in order to explain the morphological features (i.e., levees) observed in a laboratory experiment.

4.4 Comparison With Morphological Simulations and Laboratory Experiments

[66] As suggested by Rowland et al. [2010], we found that sediment eddy diffusivity depends not only on the hydrodynamics (D, B, Um) but also on the sediment characteristics (ws). Because eddy diffusivity quantifies the jet's capability to spread sediment laterally, we hypothesize that the “optimum” condition for subaqueous levees formation occurs when the eddy diffusivity is maximum, i.e., TD/TE ∼ 1.

[67] We compare our simplified model with the morphological results, using MLB as an indicator of the jet tendency to build lateral deposits. We found that for all the combinations of hydrodynamic and sedimentary conditions, the optimum lateral deposit condition is associated with TD/TE∼1 (Figures 8 and 10b). For TD/TE>>1, i.e., under mixing-limited conditions, sediments are spread over a wide fanlike area (Figure 8a). Eddies transport the sediments laterally, but the condition that focuses the deposition on the levees is not met. For TD/TE<<1, i.e., under mobility-limited conditions, eddies are not able to efficiently move sediments out of the centerline. A central deposit eventually forms where the transport capacity drops in the ZOEF (Figure 8g).

[68] In addition to the morphodynamic simulations, we perform a comparison with the laboratory experiments by Rowland et al. [2010]. These experiments, characterized by fixed hydrodynamic conditions (D=0.05 m, Um=0.55 m/s, B=0.22 m, u=13 mm/s), revealed that only a narrow range of sediment characteristics were able to form well-defined levees with no frontal bar. These sediments are characterized by a settling velocity of 10, 19, and 26 mm/s, corresponding to a depositional time scale of 2.5, 1.3, and 1 s (equation (15)). Using our simplified scaling, the eddy time scale for this experimental jet is estimated to be ∼5.7 s (equation (18)). Hence, the three experiments are characterized by TD/TE that ranges from 0.2 to 0.5, which, in accordance with our model, it is associated with high values of normalized eddy diffusivity (∼ 0.05 using Figure 11). Indeed, for such a range of eddy diffusivity, our model predicts the formation of levees (e.g., Figure 8d, which is associated with TD/TE =0.31). Discrepancies between our simulations and the experiments of Rowland et al. [2010] might be caused by the presence of slightly different depositional patterns: lunate shape in the former versus pure levee deposition in the latter.

[69] Given the uncertainty in defining both the depositional and the eddy time scales, it is possible that in a real physical setting, levee formation corresponds to values of TD/TE slightly different from one, e.g., 0.5<TD/TE<2.

4.5 Implications for Depositional Patterns at River Mouths

[70] Our simplified model assumes the presence of coherent structures, and therefore, in theory, it should be applied only to situation in which the jet is unstable. Nonetheless, a stable jet could be considered as an extreme case of the mobility-limited regime, in which there are no large vortices able to advect sediments laterally, as in the numerical experiments of Edmonds and Slingerland [2007]. As a result, a stable jet is expected to favor the formation of a frontal bar (sensu Wright [1977]).

[71] We hence conclude that unstable jets with TD/TE ∼ 1 favor the formation of lateral deposits, unstable jets with TD/TE>>1 and TD/TE<<1 favor the formation of central deposits, and that stable jets favor the formation of central deposits since they are characterized by TD/TE<<1.

[72] Clearly, this description is not exhaustive. First, our simulations are not able to build indefinitely elongated channels and the resulting lunate bar might give rise to bifurcations in the long-term evolution of the delta. It is indeed not possible to predict how the jet instability will influence the characteristic of the whole delta.

[73] Second, the transition between stable and unstable jets is far from being understood. The feedback between flow and bottom morphology is likely to affect this transition. The formation of a central deposit inhibits jet meandering, hence decreasing its eddy diffusivity and the capability to build levees. Rowland [2007] noticed that the dune field generated in the channel suppressed the meanders and hence the formation of levees. Indeed, we found a similar mechanism in some of our experiments. In the simulations with lowest velocity (simulation #1, Table 2) and a particular set of grain size (d50 equal to 96 and 144 μm), the jet becomes quickly stable after an initial deposit formation, leading to the construction of a mouth bar without significant levees. The consequence of this process is reflected in the anomalous low value of MLB (Figure 11b).

[74] The transition from meandering to stable jets, and the associated morphological feedback, should be further studied, either with numerical simulations or with laboratory experiments. In addition, the presence of waves might contribute to suppressing the meanders by increasing the effective bed friction [Soulsby et al., 1993] or to favor the instability by adding turbulence [Nardin and Fagherazzi, 2012; Nardin et al., 2013]. Tides also affect the average bed friction, increasing the spreading of the jet [Leonardi et al., 2013] and thus further stabilizing the jet.

5 Conclusions

[75] We explored the sediment dynamics of an unstable plane turbulent jet, aiming to gain insight about the depositional patterns of deltas. The flow was simulated using the software Delft3D, which numerically solves the depth-averaged shallow water equations. The jet hydrodynamics and, in particular, the fluctuating velocities and eddy viscosity, are similar to those measured in laboratory experiments [Rowland et al., 2009], suggesting that the spatial and temporal discretization used in the model is adequate to capture the unsteady components of the flow.

[76] The following results are found: (1) the sediment eddy diffusivity is a function of both hydrodynamic and sedimentary conditions, (2) the maximum sediment eddy diffusivity can be 5–10 times the eddy viscosity, which agrees with the conclusions of Rowland et al. [2010], (3) the ZOEF has higher sediment eddy diffusivity than the ZOFE, and (4) conditions that maximize the sediment eddy diffusivity lead to optimal levee growth.

[77] These results are explained by considering the coupling between the vertical dynamics of the sediments and the advection of large-scale coherent structures. The maximum eddy diffusivity, an indicator of the sediments spreading out of the jet, is reached when the time scale for particle deposition, TD, is close to the time scale for an eddy to perform one half revolution, TE. In this scenario, sediments are deposited at the farthest distance from the jet centerline and do not return back with the inward component of the eddy. The resulting morphology is characterized by the highest ratio of lateral versus central deposit, i.e., optimal conditions for levees formation. For TD/TE>>1, here defined as mixing-limited regime, sediments are transported inward and outward by the eddy, resulting in an eddy diffusivity smaller than the maximum value, as originally suggested by Rowland [2007]. For TD/TE<<1, here defined as mobility-limited regime, sediments settle before completing the eddy outward phase. The former regime is associated with fanlike central deposits, while the latter is associated with barlike central deposits.

[78] The higher sediment eddy diffusivity in the ZOEF is explained by the presence of large vortices, resulting from the interaction and merging of the smaller vortices created on the shear layers of the ZOFE. We proposed a simplified method to estimate the depositional and the eddy time scale in the ZOEF, based on settling velocity, jet width, depth, and velocity. The method is able to predict the conditions associated with the maximum eddy diffusivity and hence the optimal conditions for levees formation.

Acknowledgments

[79] We thank J. Rowland, whose insights inspired this work. This research was supported by the ACS-PRF program award 51128-ND8, by NSF award OCE-0948213 and through the VCR-LTER program award DEB 0621014, by the Office of Naval Research award N00014-10-1-0269.

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