The effects of sediment properties on deltaic processes and morphologies: A numerical modeling study

Authors


Abstract

There is a pressing need to understand how different delta morphologies arise because morphology determines a delta's ecologic structure, resilience to relative sea-level rise, and stratigraphic architecture. We use numerical modeling (Delft3D) to explain how deltaic processes and morphology are controlled by the incoming sediment properties. We conducted 36 experiments of river-dominated delta formation varying the following sediment properties of the incoming grain-size distribution: the median, standard deviation, skewness, and percent cohesive sediment, which is a function of the first three properties. Changing standard deviation and skewness produces minimal morphological variation, whereas an increase in dominant grain size (D84) and decrease in percent cohesive sediment produce a transition from elongate deltas with few channels to semicircular deltas with many channels. This transition occurs because critical shear stresses for erosion and settling velocities of grains set the number of channel mouths and the dominant delta-building process. Together, the number of channel mouths and the dominant process—channel avulsion, mouth bar growth, or levee growth—set the delta morphology. Coarse-grained, noncohesive deltas have many channels dominated by avulsion, creating semicircular planforms with relatively smooth delta fronts. Intermediate-grained deltas have many channels dominated by mouth bar growth, creating semicircular planforms with rugose delta fronts. Fine-grained, cohesive deltas have a few channels, the majority of which are dominated by levee growth, creating elongate planforms with smooth delta fronts. The process-based model presented here provides a previously lacking mechanistic understanding of the effects of sediment properties on delta channel network and planform morphology.

1 Introduction

The morphology of deltaic lobes is thought to be a function of the ratios of fluvial, wave, and tidal energies [Galloway, 1975], yet these variables do not explain the full breadth of delta morphology. For example, the Mississippi (Louisiana, USA), Mossy (Saskatchewan, Canada), and Goose (Labrador, Canada) river deltas have similar ratios of marine (wave and tidal) to fluvial energy [Syvitski and Saito, 2007], which suggests they should have similar planform morphologies (Figure 1). Instead, their morphologies range from an elongate shape with few distributary channels, to a rugose semicircle with a bifurcating channel network, to a relatively smooth semicircle with a braided channel pattern (Figure 1), indicating that additional variables control delta morphology. Previous research has suggested that grain size exerts a significant control on delta morphology [e.g., Orton and Reading, 1993]. Interestingly, these deltas are constructed by different median grain sizes, ranging from cohesive silt (Mississippi river) to noncohesive sand (Goose river) [Syvitski and Saito, 2007; D. A. Edmonds, unpublished data, 2007 and 2012]. It is difficult to truly isolate the effect of grain size in these field cases because there are other factors that vary amongst these deltas (e.g., relative sea-level change). Nonetheless, it is currently unclear if changing sediment properties alone can produce the different deltaic morphologies represented in Figure 1. To this end, the intent of this study is to address the following questions: (1) What quantitative effects do sediment properties, such as median grain size, grain-size distribution shape, and sediment cohesion, have on delta morphology? and (2) How and why is delta morphology sensitive to incoming sediment properties; that is, how do sediment properties change the dominant processes occurring on deltas? Here we attempt to answer these questions with a numerical modeling study that aims to understand the connection between sediment properties and delta morphology.

Figure 1.

Deltas with similar ratios of marine to fluvial energy (Pm:Pr) have different morphologies, ranging from (a) elongate with a few straight channels, to (b) semicircular with bifurcated channels, to (c) semicircular with braided channels. The proxy for marine power (Pm) is equal to the sum of the square of maximum-monthly wave heights and the square of the tidal range. The proxy for river power (Pr) is equal to the product of average water discharge and delta plain gradient scaled by 11 (see Syvitski and Saito [2007] for further details). Deltas are located in Louisiana, USA (Figure 1a), Saskatchewan, Canada (54° 04′ N, 102° 21′ W) (Figure 1b), and Labrador, Canada (53° 21′ N, 60° 21′ W) (Figure 1c). Data come from Syvitski and Saito [2007] (Figure 1a) and Edmonds (unpublished data, 2007 and 2012) (Figures 1b and 1c). Imagery includes USGS Landsat 7 satellite image (Figure 1a), Information Service Corporation aerial image (Figure 1b), and Google Earth aerial image ©Google Inc (Figure 1c).

2 Background on Delta Morphology

The variability and controls on delta morphology have received considerable attention, and as a starting point, we group the factors that influence delta morphology into those that originate upstream or downstream from the delta. Upstream factors control delta morphology via the fluvial system, such as river discharge [e.g., Hooke and Rohrer, 1979; Edmonds et al., 2010], channel pattern of the fluvial feeder system [e.g., Postma, 1990; Geleynse et al., 2010], and rates and properties of sediment input [e.g., Postma, 1990; Orton and Reading, 1993]. Downstream factors control delta morphology via marine processes such as salinity-induced buoyant flows, waves [e.g., Bhattacharya and Giosan, 2003; Jerolmack and Swenson, 2007; Nardin et al., 2013], tides [e.g., Dalrymple and Choi, 2007], and sea-level variations [e.g., Jerolmack, 2009].

Galloway [1975] suggested that the balance between upstream fluvial energy and downstream wave and tidal energy controls delta morphology. He claimed that fluvial energy generated by river discharge promotes basinward channel progradation, resulting in elongate deltas such as the Mississippi. Downstream marine forces generated by waves and tides work to redistribute sediment delivered to the delta apex along the adjacent shoreline. Galloway's classification was semiquantitative and only recently have studies started to show how waves and tides affect deltaic processes. In general, wave energy creates broad, asymmetric, smooth shorelines by setting up alongshore currents that redistribute sediment lateral to the shoreline and suppress mouth bar formation [Jerolmack and Swenson, 2007; Ashton and Giosan, 2011; Nardin and Fagherazzi, 2012]. Additionally, wave energy has been shown to create shoreline perturbations, such as spits [Ashton and Giosan, 2011], and influence mouth bar formation patterns [Nardin and Fagherazzi, 2012; Nardin et al., 2013] due to varying degrees of wave amplitude and angle to the shoreline. Tides cause the remobilization of sediment, eroding mouth bars and maintaining abandoned distributary channels, which results in delta morphologies characterized by dendritic channel networks with a large number of distributaries that widen basinward [Dalrymple and Choi, 2007; Fagherazzi and Overeem, 2007; Fagherazzi, 2008; Leonardi et al., 2013].

While the classification by Galloway [1975] accounts for first-order differences in delta morphology, it does not explain the full variability (e.g., Figure 1). Orton and Reading [1993] explored the effects of sediment properties and collected a convincing data set that shows deltas constructed from larger median grain sizes are semicircular, whereas fine-grained deltas are more irregular in planform. Fine-grained, elongate deltas are characterized by large, stable distributaries with low gradients and straight to sinuous patterns. In contrast, coarse-grained deltas have a larger number of small, ephemeral distributaries with a higher gradient and a braided pattern [McPherson et al., 1987; Orton and Reading, 1993]. Although the field data are robust, it is not obvious how different sediment properties modify delta-building processes and produce different morphologies.

The next logical step is to explore, from a process-based perspective, why and how the link between delta morphology and sediment properties exists. Recent progress has been made through the use of physical [Hoyal and Sheets, 2009; Martin et al., 2009] and numerical [Edmonds and Slingerland, 2010; Geleynse et al., 2011] experiments of delta formation. Edmonds and Slingerland [2010] investigated how the sediment cohesion of fine grains controls delta morphodynamics by changing ratios of cohesive to noncohesive sediment entering the system and values of critical shear stress for erosion of the cohesive sediment. More cohesive sediment builds stronger levees, allowing channels to prograde basinward and produce rugose shorelines. In contrast, less cohesive conditions build weaker levees that allow relatively unconfined channels to distribute sediment across the entire delta plain, producing semicircular deltas with smoother shorelines. Geleynse et al. [2011] used numerical modeling to show that subsurface sediment composition controls delta morphology in a similar way. Higher ratios of fine, cohesive sediment present in the initial basin subsurface produce elongate features with incisive distributaries independent of grain-size ratios entering the system. Furthermore, this effect is evident in the presence of both waves and tides, suggesting sediment properties may exert a stronger control than previously thought.

Given that previous research suggests the effect of sediment properties on delta morphology may be a first-order control, there is a pressing need to understand this link. To this end, we conducted 36 numerical experiments of delta growth under varying incoming sediment properties. We altered the grain-size distribution by changing the median grain size, standard deviation, and skewness. Different grain-size distributions may have different percentages of incoming sediment that is cohesive, because grains ≤ 64 µm are considered cohesive in the model.

This study differs from similar recent studies [Edmonds and Slingerland, 2010; Geleynse et al., 2011] in a few key ways. The explicit aim here is to explore how the grain-size distribution, rather than just sediment cohesion, controls deltaic processes and morphology. We focus on grain-size distribution because that is a variable that is easily constrained and measured in the field. The previous studies focused primarily on the effects of cohesive sediments, and the grain-size distributions in the models were represented by only one or two grain sizes. Furthermore, the past studies did not consider the full range of cohesive sediment input, but here we systematically consider the variation from coarse-grained, noncohesive deltas to fine-grained, cohesive ones in one study.

3 Methods

3.1 Description of the Delft3D Model

We model deltaic processes and formation using Delft3D, which is a physics-based morphodynamic model that simulates flow and sediment transport. The program has been used for various hydrodynamic and sediment transport studies [e.g., Lesser et al., 2004; Edmonds and Slingerland, 2007; Van Maren, 2007; Edmonds and Slingerland, 2008; Hu et al., 2009; Van Maren et al., 2009] including models of delta growth [Marciano et al., 2005; Dastgheib et al., 2008; Edmonds and Slingerland, 2010; Geleynse et al., 2010, 2011]. Flow is computed by solving the depth-integrated, Reynolds-averaged Navier-Stokes equations for incompressible and free surface flow. Hydrodynamic results are then used to compute suspended and bedload transport, and the bed elevation is subsequently updated according to divergences in sediment transport. The remaining discussion of Delft3D will focus on the mathematical treatment of sediment transport for cohesive and noncohesive grains since that is most relevant to this study. For further description of hydrodynamic calculations, see Deltares [2013].

Cohesive and noncohesive sediment transport, erosion, and deposition are handled separately in Delft3D. Any sediment fraction ≤ 64 µm in diameter is considered cohesive sediment in the model, whereas those > 64 µm in diameter are noncohesive. Cohesive sediment is only transported in suspension, whereas noncohesive sediment is transported as both suspended load and bedload. Furthermore, cohesive sediment is subject to user-defined critical shear stresses for erosion and deposition.

Transport of suspended sediment (both cohesive and noncohesive) is computed by solving the depth-averaged version of the 3-D advection-diffusion equation:

display math(1)

where ci is the mass concentration of the ith sediment fraction (kg m−3) assuming a standard Rouse profile concentration gradient, ux, uy, and uz are the x-, y-, and z-directed fluid velocities (m s−1), respectively, ws,i is the settling velocity of the ith sediment fraction (m s−1), and εs,x,i, εs,y,i, and εs,z,i are directional eddy diffusivities of the ith sediment fraction (m2 s−1). Settling velocities of cohesive sediment fractions are set according to Stokes’ law, and the effects of cohesive sediment flocculation are ignored. Noncohesive sediment settling velocities of the ith sediment fraction (ws,i) are calculated according to Van Rijn [1993] depending on the user-defined grain diameter, such that

display math(2)

where R = ρs / ρw − 1 is the submerged specific gravity, ρs is the specific density of sediment (kg m−3), ρw is the specific density of water (kg m−3), g is acceleration due to gravity (9.8 m s−2), Di is the grain diameter of the ith sediment fraction (m), and ν is the kinematic viscosity coefficient of water (m2 s−1).

Erosive and depositional fluxes of suspended sediment are computed separately for noncohesive and cohesive sediment. Exchange of noncohesive suspended sediment with the bed is calculated as an erosive flux due to upward diffusion and depositional flux due to sediment settling. Erosion and deposition of cohesive sediment are calculated according to the Partheniades-Krone formulations [Partheniades, 1965]:

display math(3)

where Fe,i  and Fd,i are the erosive and depositional fluxes, respectively, of the ith cohesive sediment fraction (kg m−2 s−1), τ0 is bed shear stress (N m−2); τce(C) and τcd(C) are user-defined critical shear stresses for erosion and deposition, respectively, of cohesive sediment, and cb,i; is the sediment concentration near the bed of the ith sediment fraction (kg m−3). These fluxes are then applied as source and sink terms and the bed level is updated accordingly.

The transport of bedload is calculated by the method described by Van Rijn [1993]:

display math(4)

where qb,i is bedload sediment discharge per unit width of the ith sediment fraction (m2 s−1), u is depth-averaged velocity (m s−1), and uc,i is the critical depth-averaged velocity (m s−1) for initiation of motion of the ith sediment fraction based on the Shields curve. The direction of bedload transport is determined by local flow conditions and is adjusted for bed-slope effects [Bagnold, 1966; Ikeda, 1982]. While suspended load transport entering the upstream open boundary is prescribed as a boundary condition, the bedload transport is always in equilibrium with the local hydrodynamic conditions. Thus, time-averaged bedload sediment discharges at the upstream boundary vary between simulations.

We employ a version of Delft3D that preserves the subsurface stratigraphy by tracking the deposited grain sizes as stacked bed layers. Only grains in the topmost bed layer are available for erosion. Thus, the spatial variation in erosion due to the presence of mud or sand in the topmost bed layer is taken into account.

3.2 Model Setup

Our experimental setup is based on the runs by Edmonds and Slingerland [2010], and we list all user-defined model parameters in Table 1. The 2-D depth-averaged model is computed on a grid of 300 × 225 computational cells, with each cell 25 × 25 m, which creates a 7500 × 5625 m basin. We model a river entering a water body in the absence of waves, tides, base-level change, and salinity. The effects of salinity-induced buoyant flows and cohesive sediment flocculation are not considered.

Table 1. User-Defined Model Parameters for Runs in This Study
User-Defined Model ParameterValueUnits
Grid size300 × 225cells
Cell size25 × 25m
Initial basin bed slope0.000375
Initial channel dimensions (width × depth)250 × 2.5m
Upstream open boundary: incoming water discharge1000m3 s−1
Downstream open boundary: constant water surface elevation0m
Initial sediment layer thickness at bed10m
Subsurface stratigraphy bed layer thickness0.1m
Number of subsurface stratigraphy bed layers75
Time step0.2min
Morphological scale factor175
Spin-up interval before morphological updating begins720min
Spatially constant Chézy value for hydrodynamic roughness45m1/2 s−1
Background horizontal eddy viscosity and diffusivity (added to subgrid horizontal large eddy simulation)0.0001m2 s−1
Factor for erosion of adjacent dry cells0.33
Number of sediment fractions7
Cohesive sediment critical shear stress for erosion (τce(C))1N m−2
Cohesive sediment critical shear stress for deposition (τcd(C))1000N m−2

The initial conditions of our domain consist of a basin floor with a slope of 0.000375 to the north (Figure 2) and an initial bed roughness with amplitudes of a few centimeters imposed on the basin floor. At the southern boundary, the initial channel is 250 m wide, 2.5 m deep, and 500 m long, cut through an erodible beach. The beach has an elevation high enough that all water is confined to the channel throughout each simulation. Everywhere there is 10 m of erodible sediment in the subsurface that is homogeneously mixed in the same proportion to the incoming grain-size distribution for each run.

Figure 2.

Numerical model setup in plan view. Model domain consists of grid cells that are 25 × 25 m in size. Colors represent initial bathymetry. Black arrow indicates direction of incoming flow. Wide black border lines indicate open boundaries of constant water surface elevation at 0 m.

We place open boundaries of constant water surface elevation along the north, east, and west boundaries of the grid (wide black lines in Figure 2) and specify an incoming water and sediment discharge at the channel on the southern boundary. The water discharge of Q = 1000 m3 s−1 entering the basin is temporally constant and carries a noncohesive bedload discharge in equilibrium with the local hydrodynamic conditions and a user-defined discharge of cohesive and noncohesive suspended load. The incoming suspended sediment discharge for cohesive and noncohesive sediment totals 0.0377 m3 s−1. This number was chosen because it is representative for deltas of the world [Syvitski and Saito, 2007]. As previously described, bedload sediment discharge entering the system is a product of time-varying hydrodynamic conditions at the upstream open boundary. As a result, time-averaged total sediment discharges (inline image) vary among runs (Table 2).

Table 2. Model Run IDs and Grain Size Boundary Conditions for This Study
Run IDD50 (mm)φ50 (φ)σ (φ)SkD84 (mm)φ84 (φ)% Cohesiveinline image (m3 s−1)
A1a10.016.60.100.0116.5461000.0377
B2a10.016.610.70.0146.1611000.0377
B1a10.016.6100.0195.71898.130.0377
B3a10.016.61−0.70.0295.11597.360.0378
C1a10.016.6200.0384.72196.290.0378
D1a10.016.6300.0693.85371.770.0384
A1c10.054.30.100.0544.2241000.0377
B2c10.054.310.70.0713.82378.030.0378
B1c10.054.3100.0963.38473.350.0379
B3c10.054.31−0.70.1422.81453.510.0378
B5c10.054.31−0.350.1282.96461.390.0384
C1c10.054.3200.1822.45972.850.0381
D1c10.054.3300.3451.53771.070.0386
B2d10.0753.710.70.1053.24644.510.0383
A1e10.13.30.100.1073.22300.0441
B2e10.13.310.70.1422.81844.300.0384
B1e10.13.3100.1912.38526.170.0395
B3e10.13.31−0.70.2741.86720.260.0411
B4e10.13.310.350.1582.66238.060.0380
C1e10.13.3200.3681.44224.360.0398
D1e10.13.3300.7190.47623.370.0421
A1h10.252.00.100.2671.90700.0483
B2h10.252.010.70.3591.4767.910.0441
B1h10.252.0100.4801.0603.280.0447
B3h10.252.01−0.70.7100.49500.0493
C1h10.252.0200.8970.15724.520.0390
D1h10.252.0301.769−0.82319.920.0610
A1m10.51.00.100.5330.90700.0509
B2m10.51.010.70.7130.4881.640.0479
B1m10.51.0100.9810.0280.040.0485
B3m10.51.01−0.71.400−0.48500.0556
C1m10.51.0201.826−0.8692.940.0433
D1m10.51.0303.527−1.81822.940.0383
A1o110.00.101.070−0.09800.0492
B1o110.0101.896−0.92300.0478
C1o110.0204.063−2.0231.660.0438

For each model run, the incoming sediment discharge is partitioned into seven different sediment fractions within Delft3D, where a sediment fraction represents a given grain size (Figure 3a). Each sediment fraction has a discharge at the boundary in proportion to its weight frequency within the grain-size distribution. Sensitivity tests varying the number of sediment fractions showed that a grain-size distribution discretized by seven sediment fractions results in the same overall delta shape, shoreline characteristics, and channel network as a higher resolution grain-size distribution of 14 sediment fractions.

Figure 3.

Example grain-size distributions showing (a) discretization of a continuous grain-size distribution into seven sediment fractions represented as a histogram (example shown for D50 = 0.1 mm, σ = 2 φ, Sk = 0), (b) the range in median grain sizes for a given σ and Sk (examples shown for σ = 1 φ, Sk = 0), (c) the range in standard deviations for a given D50 and Sk (examples shown for D50 = 0.1 mm, Sk = 0), and (d) the range in skewnesses for a given D50 and σ (examples shown for D50 = 0.1 mm, σ = 1 φ). The vertical black line in Figures 3b–3d represents the cohesive/noncohesive grain size threshold (0.064 mm).

Delft3D requires the user to define values for critical shear stress for erosion (τce(C)) and deposition (τcd(C)) of the cohesive sediment fractions (equation (3)). Erosion of mud occurs when the applied stress exceeds τce(C) = 1 N m−2, which represents a mud of intermediate cohesion [Black et al., 2002]. We specify continuous deposition of mud (by setting τcd(C) = 1000 N m−2) that is balanced by an erosive flux, such that equilibrium depth occurs when the two fluxes are equal.

Each model run (Table 2) is computed until an equal amount of sediment has entered the domain, and we compare runs once they reach a state of dynamic equilibrium (defined in section 4). To speed up the computations, we use a morphological bed updating factor of 175. Assuming that rivers experience bankfull (i.e., geomorphically effective) conditions ~10 days per year, these simulations represent ~290 years of change. Thus, the results of these experiments are meant to represent individual delta lobes that grow over small temporal and spatial scales, whereas larger deltaic systems may include multiple lobes dominated by various controls [e.g., Bhattacharya and Giosan, 2003].

We conducted a suite of test runs to assess the sensitivity of the results to user-defined model parameters. The tested parameters include grid resolution, number of sediment fractions, time step, morphological bed updating factor, subsurface bed layer thickness, bedload transport formulation, and roughness formulation. Changing these parameters indeed created different deltas, but they had minimal effects on topset gradient, number of channel mouths, delta front rugosity, and delta shape, which are morphometric parameters of interest for this study.

3.3 Parameter Space

The parameter space explored in this study is defined by changes in the grain-size distribution and percent cohesive sediment of the incoming sediment load at the delta apex. Although there is a small variation (less than a factor of 2) in sediment discharge among runs due to the equilibrium bedload discharge boundary condition, it has little effect on delta morphology, as we discuss later. We vary four properties of the grain-size distribution (Table 2 and Figures 3b–3d): (1) median grain size (D50), (2) standard deviation (σ), which characterizes the sorting of the sediment input, (3) skewness (Sk), which determines whether the bulk of the sediment is larger or smaller in size than the mean grain diameter, and (4) percent cohesive sediment, which varies as a function of the percentage of fine grains (≤ 64 µm) in the grain-size distribution and determines the bulk amount of cohesion within the system.

In this study we model the incoming grain-size distribution as a unimodal, log-normal distribution (normal in phi space), though there is debate as to whether a bimodal or even a log-hyperbolic distribution is more common in natural river systems [e.g., Hajek et al., 2010]. We use a unimodal distribution because it is a simpler distribution characterized by fewer parameters (i.e., one median, standard deviation, and skewness). This makes constraining cause and effect between the input grain-size distribution parameters and the output delta morphology metrics more straightforward.

The grain-size distribution parameters are calculated in phi values (φ), defined as

display math(5)

where φi is the phi value of the ith sediment fraction and Di is the grain size measured in millimeters. In our runs we vary median grain size from ~6.64 φ to 0 φ, (Figure 3b), which corresponds to changing D50 from 0.01 mm (silt) to 1 mm (coarse sand), consistent with global deltaic systems [Orton and Reading, 1993; Syvitski and Saito, 2007]. Standard deviation (σ) is calculated by the following formula [Folk, 1974]:

display math(6)

where fi is the weight percent of each ith sediment fraction. Standard deviation of the distribution is varied from 0.1 φ (well sorted) to 3 φ (poorly sorted) (Figure 3c). Skewness (Sk) is calculated after Folk [1974]:

display math(7)

and is varied from −0.7 to 0.7, which encompasses varying degrees of fine- and coarse-skewed distributions as well as a normal Gaussian distribution (Sk = 0) (Figure 3d). For example, a positively skewed distribution corresponds to a tail that lies to the right of the distribution's mean, or the finer grain size end. It follows that a positively skewed distribution will be composed of a higher volume of sediment below the cohesive grain size threshold and thus represents a more cohesive sediment input. Similarly, a negatively skewed distribution represents a sediment input characterized by a higher volume of larger grain sizes and thus a less cohesive sediment load.

Because grains ≤ 64 µm are defined as cohesive, variations in the above mentioned grain-size distribution parameters determine the fraction of cohesive sediment (Figures 3b–3d and Table 2). We select grain-size distributions where the percent cohesive sediment ranges from 0 to 100%, which is a larger range than that considered by Edmonds and Slingerland [2010]. Although additional variables affect bulk cohesion within natural systems (e.g., vegetation, wetting and drying of deposits), here we only consider the effect due to the mud to sand ratio. Because grain size and cohesiveness of sediment are inextricably linked in both the model and natural systems, it is difficult to isolate the two, though in section 5 we attempt to elucidate the separate controls of each sediment property.

4 Results

Based on 36 runs of delta growth, each with different sediment properties (Table 2 and Figures 4 and 5), we observe that as grain sizes increase, deltas undergo a morphological transition from elongate in planform with a few stable channels to semicircular in planform with many mobile channels (Figure 4). We find that variations in the standard deviation and skewness of the grain-size distribution have minimal effects on delta morphology because they do not produce large enough variations in D84 and percent cohesive sediment—the sediment properties we find exert the dominant control on delta morphology.

Figure 4.

Modeled deltas display different morphologies based on variations in median grain size (D50), whereas the effect of standard deviation (σ) is more subtle. Deltas are pictured after the same volume of sediment has entered the system. Note that both x and y axis intervals vary. For all runs, Sk = 0.

Figure 5.

Modeled deltas display little morphological variation based on changing skewness of the grain-size distribution. Deltas are pictured when the same volume of sediment has entered the system. Note that y axis intervals vary. For all runs, σ = 1 φ.

To quantify the observed morphological transition, we measure four morphometric parameters on each delta: topset gradient, number of channel mouths, delta front rugosity, and delta shape (Figure 6). All simulations are compared once the delta has reached a state of dynamic equilibrium. Dynamic equilibrium is achieved when topset gradient, number of channel mouths, and delta front rugosity are roughly constant through time. In our runs this usually occurs by time t ~ 0.6 T (Figure 7), where t is current time in delta growth and T is total time of delta growth. All reported metrics, except for delta shape, are averaged over the final one third of the run. Delta shape does not achieve a dynamic equilibrium because elongate deltas become more elongate through time.

Figure 6.

Model outputs depicting how (a) delta front rugosity (Run ID = B1c1, rugosity value ≈ 1.47) and (b) delta shape, equation (8) (Run ID = C1m1, A ≈ 0.90), are measured. See sections 4.3 and 4.4 for description of calculations.

Figure 7.

Morphometric parameters reach a dynamic equilibrium during delta growth. (a) Average topset gradient becomes stable at ~0.5 t/T. (b) Average number of channel mouths becomes relatively stable at ~0.6 t/T. (c) Delta front rugosity becomes stable at ~0.5 t/T. All examples shown have σ = 0.1 φ, Sk = 0.

In our analyses we characterize the grain-size distributions by their D84 and percent cohesive sediment (Table 2). We use D84 because it is sensitive to changes in D50, σ, and Sk of the distribution. We plot all regressions using a negative phi value (− φ84 = log2D84), where D84 is in millimeters. We define phi as negative, rather than the standard positive phi, so that grain size increases to the right on bivariate plots. All regressions shown are significant at the PN = 0.05 level (following Taylor [1997]), unless otherwise noted.

4.1 Topset Gradient

To calculate these metrics, we first classify all grid cells in the modeling domain as shoreline, open water, channelized topset, or land. We use the Opening-Angle Method [Shaw et al., 2008] to delineate the delta shoreline. This method defines a grid cell as “open water” if from that grid cell there is an angular swath “open” to the ocean (i.e., a view not obstructed by land) greater than some threshold. If the angular swath is less than the threshold, it is classified as “land.” We use an angular swath threshold of 70°. The shoreline is then defined as the boundary between the “open water” and “land” domains. This permits an objective definition of the shoreline because it is otherwise difficult to decide if, for example, a detached mouth bar is part of the shoreline or how to draw the shoreline across channel mouths.

To calculate delta topset gradient, we measure rays from the delta apex to the shoreline. The elevation of the delta apex is defined by averaging all water surface elevations within a 500 m swath around the head of the delta. The shoreline is discretized into points spaced ~20 m apart. Given that the deltas are small and backwater length scales are usually long [e.g., Lamb et al., 2012], we assume there are no strong concavities in the water surface slope and assign a linear slope between the apex elevation and shoreline points. A representative topset gradient is found by averaging all gradients in space over the period of dynamic equilibrium.

Average measured topset gradients increase as D84 increases (R2 = 0.73) (Figure 8a). Topset gradient varies from 1 × 10−4 to 2 × 10−4 for the finest-grained deltas (D84 = 0.01–0.02 mm) and 6 × 10−4 to 1 × 10−3 for the coarsest-grained deltas (D84 = 2–4 mm). Topset gradient and percent cohesive sediment exhibit an inverse relationship (Figure 8e) (R2 = 0.36), given that an increase in percent cohesive sediment is achieved by decreasing median grain size.

Figure 8.

Measured morphometric parameters are correlated with changes in (a–d) D84 and (e–h) percent cohesive sediment input. An increase in D84 and a decrease in percent cohesive sediment result in steeper topset gradients (Figures 8a and 8e), more channel mouths (Figures 8b and 8f), and more semicircular delta shapes (equation (8); Figures 8d and 8h). A weak parabolic relationship exists between delta front rugosity and D84 (Figure 8c) and percent cohesive sediment (Figure 8g), where the most rugose delta fronts result from intermediate D84 and percent cohesive sediment input.

4.2 Number of Channel Mouths

For all the grid cells marked as “land,” we differentiate between those that are channelized and those that are unchannelized. Active channelized cells are defined by depth (h), water velocity (u), and total sediment discharge (Qt) thresholds, where a cell is considered channelized if h ≥ 0.25 m, u ≥ 0.2 m s−1, and Qt ≥ 2.25 × 10−4 m3 s−1 (per grid cell). All remaining cells within the shoreline polygon that are not channelized are considered land. A channel mouth is defined as a location where two or more adjacent channelized cells intersect the shoreline.

The average number of active channel mouths at dynamic equilibrium increases as D84 increases (R2 = 0.77) (Figure 8b). Finest-grained deltas (D84 = 0.01–0.02 mm) average 1–4 active channel mouths, whereas coarser-grained deltas (D84 = 2–4 mm) have 9–14 active channel mouths. The average number of active channel mouths decreases as percent cohesive sediment increases (R2 = 0.58) (Figure 8f), though the relationship has more scatter.

4.3 Delta Front Rugosity

We measure delta front rugosity because we aim to capture the effects of delta front deposition (e.g., subaqueous mouth bars). Delta front rugosity is defined using the elevation contour at 1 m water depth. Sensitivity tests show that there is minimal difference in computed delta front rugosity values at different contour elevations.

We calculate delta front rugosity as a sinuosity value where the total delta front length is normalized by the length of a smoothed delta front location (Figure 6a). A large sinuosity measurement represents a very rugose delta front characterized by larger-scale deviations. The smoothed delta front location is computed using a moving average window with a length of 50 delta front points (each located ~20 m apart). Sensitivity tests show variations in moving average window length produce little variation in measured relationships between delta front rugosity and sediment properties.

Delta front rugosity exhibits a weak parabolic relationship with respect to D84 (R2 = 0.22) (Figure 8c) and percent cohesive sediment (R2 = 0.46) (Figure 8g). The most rugose delta fronts, with values of ~1.45–1.55, are created by intermediate grain sizes. Rugosity values decrease to ~1.3 for the finest grain sizes (D84 = 0.01–0.02 mm) and ~1.3–1.5 for the coarsest grain sizes (D84 = 2–4 mm). These relationships suggest that cohesive, fine-grained deltas exhibit the smoothest delta fronts and intermediate-grained deltas exhibit the most rugose delta fronts, while noncohesive coarse-grained deltas have moderately smooth delta fronts.

4.4 Delta Shape

Delta shape is quantified with a simple metric:

display math(8)

where A is delta shape, B is delta width defined as the maximum beach-parallel distance across the delta shoreline, and L is delta length defined as the maximum beach-perpendicular distance (Figure 6b), where “beach” is the southern boundary of the model domain (Figure 2). Deltas with A = 1 are semicircular, deltas with A < 1 are elongate with long axes perpendicular to the beach, and deltas with A > 1 have long axes that are beach parallel. While A reaches a dynamic equilibrium for semicircular deltas, elongate deltas will continue to grow causing A to decrease with time. To obtain a representative value of A, delta shape is measured when a predetermined amount of sediment has entered the system.

Delta shape (A) increases as D84 increases (R2 = 0.58) (Figure 8d) and decreases as percent cohesive sediment increases (R2 = 0.40) (Figure 8h). Fine-grained, cohesive deltas (D84 = 0.01–0.04 mm) exhibit delta shape values ranging from A = 0.3 to 0.6, indicating that their planform morphologies are elongate and deviate from a semicircle. Coarser-grained deltas have delta shape values A ≈ 1 and thus approximate a semicircle.

Interestingly, some of the coarsest-grained deltas have values A > 1 which suggests there is a preference of the delta shoreline to prograde in the beach-parallel direction rather than basinward. This is likely due to the shallower depths that exist along the initial shoreline, creating higher rates of progradation per unit volume of sediment deposition relative to deeper locations.

5 Discussion

As detailed in the previous section, we observe a morphological transition in deltas as grain size increases and percent cohesive sediment decreases. In this section, we explore how sediment properties create the observed variations in delta morphology. As a starting point to frame our discussion, we note that changing the dominant grain size and percent cohesive sediment input alter two important characteristics of the sediment constructing a delta—the critical shear stress for erosion (τce) and the settling velocity of grains in transport (ws). The relationship between grain size and ws is straightforward—as grain size increases, ws increases nonlinearly. However, the relationship between grain size and τce is more complex because fine grains are inherently cohesive. In general, the coarsest grains have high τce values, the intermediate-sized grains have low τce values, and the fine, cohesive grains (≤ 64 µm in this study) can have the highest τce values.

5.1 How Do Sediment Properties Control the Number of Channel Mouths?

5.1.1 Grain Size Sets Topset Gradient

Our results show that an increase in grain size creates a delta with a steeper topset gradient (Figure 8a). This relationship arises because a delta increases its topset slope to create higher bed shear stresses in order to transport larger grain sizes with higher τce values, as has been well established for alluvial fans [Parker et al., 1998; Whipple et al., 1998]. By similar reasoning, one might expect to see an increase in topset gradient of fine-grained, cohesive deltas due to the higher τce of cohesive sediment. Instead, we find that an increase in percent cohesive sediment leads to shallower topset gradients (Figure 8e). This occurs because the ws of fine, cohesive grains is slow, and these grains bypass the delta and are advected into the basin. Indeed, we observe that the fine-grained, cohesive deltas have the most sediment deposited on the delta bottomset. Sediment discharge (Qt) to water discharge ratios have also been shown to set the gradient on alluvial fans [Parker et al., 1998; Whipple et al., 1998], though as Qt varied only slightly amongst runs (Table 2), the relationship between Qt and gradient in this study is weak (R2 = 0.15) (data not shown here).

5.1.2 Topset Gradient and Percent Cohesive Sediment Set Channel Time Scale

Deltas with steeper topset gradients should have more mobile channels that frequently shift course, which has been shown in fluvial [Ashworth et al., 2004] and alluvial fan systems [Bryant et al., 1995]. To quantify channel mobility on the delta topset, we calculate an average channel time scale inline image (years) (Figure 9a). Strictly speaking, our inline image represents the average length of time a channel stays in a given position on the delta topset and does not differentiate between lateral mobility from bank erosion or avulsion.

Figure 9.

(a) The ratios of channelized grid cells that remain active after some time (Gch) (solid lines) are fit with decay curves (dashed lines) to measure channel abandonment rate. Complete abandonment is defined as the abandonment of 90% of the initially active channel cells, or Gch = 0.1 (horizontal dotted line). Examples show the average channel abandonment pattern for three deltas. Note that the time axis ends at 3 years for figure scaling, whereas channel lifetime may be longer. (b) Good agreement between the theoretical predicted avulsion time scale TA (equation (10)) and inline image indicates channel mobility on these deltas is dominated by avulsions. TA values most accurately predict inline image values when ψ ≈ 0.74 (equation (10)), as shown by the exponent of the power law relationship (R2 = 0.72).

We calculate inline image by tracking channel initiation and subsequent abandonment through time. This is done by calculating the mean lifetime of the channelized grid cells by fitting an exponential decay function to their abandonment (dashed lines in Figure 9a). Similar to the method used by Wickert et al. [2013], we use the equation

display math(9)

where Gch is the ratio of channelized grid cells that remain active after channel initiation, λ is the decay constant, and tch is some time after channel initiation. Mean channelized grid cell lifetime is then found by inverting λ. The decrease in Gch is calculated as channelized grid cells become inactive (defined as no longer meeting the active channelization threshold requirements). We truncate the distribution at Gch = 0.1 (dotted line in Figure 9a) to avoid influence from outlier channelized grid cells that remain active throughout delta growth. It follows that a delta with a rapid decline in Gch (gray lines in Figure 9a) has channelized grid cells that are abandoned quickly after formation. Deltas with Gch values that decline slowly (wide black lines in Figure 9a) have relatively stable channelized cells. inline image is then calculated by scaling λ−1 based on the channel width in number of channelized grid cells. We use this method, instead of directly measuring individual channel lifetimes, because channels in some deltas persist for the entire run and this allows us to estimate channel lifetime beyond the length of the model run.

The average channel time scale (inline image) in our numerical experiments decreases with increasing D84 (R2 = 0.71) (Figure 10a) and increases with increasing percent cohesive sediment input (R2 = 0.74) (data not shown here). These relationships demonstrate that less cohesive, coarser-grained deltas have more laterally mobile channels, whereas more cohesive, finer-grained deltas have stable channels. This is consistent with other work that has also shown an increase in fine, cohesive sediment leads to more stable channels in both deltaic [Hoyal and Sheets, 2009; Martin et al., 2009; Edmonds and Slingerland, 2010] and fluvial [e.g., Van Dijk et al., 2013] systems. This relationship is due to the higher τce values of the cohesive sediment deposited in channel banks which makes them more resistant to erosion [Hoyal and Sheets, 2009].

Figure 10.

(a) Coarser-grained deltas have channel mouths with shorter channel time scales (inline image) created by steeper topset gradients and decreased percent cohesive sediment, and (b) shorter channel time scales (higher avulsion frequencies) result in more channel mouths. (c) A nonmonotonic relationship exists between the percentage of channels with Trmb/inline image < 1 and D84 with a peak at intermediate-grained sediment inputs, (d) which relates to increased delta front rugosity created by increased mouth bar growth and progradation. (e) A decrease in channel mouth width to depth ratios (empty circles) created by smaller D84 and related higher percent cohesive sediment (not shown) creates deltas with increased percentages of channel mouths with unstable turbulent jets (black circles). (f) Deltas with high percentages of channel mouths with unstable jets have elongate delta shapes (equation (8)) because these jets create prograding subaqueous levees which maintain elongated channels. R2 values shown do not include outliers. Inclusion of outliers produces R2 = 0.40 (Figure 10a) and R2 = 0.31 (Figure 10b).

We hypothesize that the magnitude of inline image is dominantly set by avulsion dynamics. The majority of aggradation in fluvial systems occurs in or near channels [Pizzuto, 1987; Heller and Paola, 1996; Törnqvist and Bridge, 2002], which leads to superelevation of the channel relative to the adjacent floodplain and subsequent channel avulsion [Bryant et al., 1995; Heller and Paola, 1996; Mohrig et al., 2000]. A theoretical channel avulsion time scale, TA (years), may be predicted as [Jerolmack and Mohrig, 2007]

display math(10)

where inline image is average channel depth (m) and inline image is channel aggradation rate (m yr−1). ψ is the threshold channel superelevation normalized to channel depth required for avulsion. Using equation (10), we calculate theoretical channel avulsion time scales (TA) for all runs by measuring the spatially averaged in-channel aggradation rates and channel depths near the delta apex.

TA values are similar to our measured inline image values (R2 = 0.72) (Figure 9b), which suggests that inline image is set by aggradation-driven channel avulsion. Theoretical TA values most accurately predict measured inline image values when ψ ≈ 0.74 (Figure 9b), suggesting channels in this study are abandoned before they aggrade an entire channel depth. Although this value differs from that used in the study of Jerolmack and Mohrig [2007] (ψ = 1), it falls well within the range of values published by Mohrig et al. [2000], who showed both modern and ancient systems often avulse before achieving a relative superelevation equal to an entire channel depth. We interpret the relationship between inline image and grain size to be the result of faster aggradation rates, which create more frequent avulsions on coarse-grained deltas according to equation (10).

5.1.3 Channel Time Scale Sets the Number of Channel Mouths

Our results show that deltas with shorter inline image values also have a larger number of channel mouths (Figure 10b). We suggest that increased avulsion frequency on coarse-grained deltas, driven by steeper slopes and low sediment cohesion, leads to an increase in number of channel mouths on coarse-grained deltas. In our model runs, we observe three ways that more frequent channel avulsions create more channels. First, avulsions in our runs are often partial and do not always completely abandon the parent channel. Therefore, an avulsion creates two channels from one parent channel. Second, frequent avulsions increase the frequency of overbank flow. The overbank flow is often erosive, especially for steep coarse-grained deltas, and creates channels via incision [Slingerland and Smith, 2004; Hajek and Edmonds, 2014]. Third, frequent avulsions leave behind relict channel pathways. Because the avulsion rate on coarse-grained deltas is high, the relict channels on the topset are not annealed and are frequently reoccupied during periods of overbank flow [Reitz et al., 2010]. If enough fine grains are present to fill the relict channels, then this effect may not be present.

Deltas with longer inline image values have fewer channel mouths (Figure 10b). In general, these shallow, fine-grained cohesive deltas have highly stable channels [Hoyal and Sheets, 2009; Martin et al., 2009; Edmonds and Slingerland, 2010] that rarely avulse or bifurcate around mouth bars, resulting in a decreased number of channel mouths at the delta shoreline (Figures 8b and 8f). Previous research [Edmonds and Slingerland, 2010] established a link between cohesion and number of bifurcations. Edmonds and Slingerland [2010] considered sediment mixtures ranging from ~55% to 96% cohesive sediment and found a peak in the number of bifurcations at intermediate cohesion. While we did not strictly measure the number of bifurcations, the number of channel mouths should be related to the number of bifurcations provided channels are generally distributive and do not rejoin downstream from the bifurcations frequently. For deltas with dominantly cohesive sediment inputs (~50%–100% cohesive sediment), the results of this study recover a similar nonmonotonic relationship between the number of channel mouths and percent cohesive sediment as reported by Edmonds and Slingerland [2010] (dashed line in Figure 8f). The peak in this relationship occurs when the sediment input is ~70%–80% cohesive. This increase in number of channel mouths (7–12) is attributed to increased stability of mouth bars by higher τce values of cohesive sediment, which leads to more frequent channel bifurcations around stable mouth bars. A further increase in percent cohesive sediment (~80%–100%) leads to a decrease in the number of channel mouths, because the increased resistance of levees inhibits both avulsions and mouth bar growth [Edmonds and Slingerland, 2010].

5.2 How Do Sediment Properties Control Delta Planform Morphology?

In the previous section we explained how sediment properties control the number of channel mouths by setting the topset gradient (Figure 8a) and channel time scale (Figure 10a and 10b). We now suggest that a delta's planform shape is set by both the number of channel mouths at the shoreline and the dominant process controlling their behavior. At the risk of oversimplification, we suggest that a given distributary channel mouth can behave in one of three ways: avulse to a new location, bifurcate around a mouth bar, or prograde via levee elongation (assuming the right balance of sediment supply and accommodation). The number of channel mouths is important in that it dictates the spatial distribution of the effects of the above behaviors (e.g., mouth bar growth) along the delta shoreline.

5.2.1 How Do Sediment Properties Control Delta Front Rugosity?

We find that delta front rugosity values are greatest for intermediate-grained deltas, whereas coarser-grained deltas have less rugose delta fronts, and finest-grained deltas have the smoothest delta fronts (Figure 8c). Observed delta front rugosity is created by differential progradation rates of the delta front, which in most cases corresponds to zones of mouth bar deposition. Thus, we would expect the most rugose delta fronts to be created by deltas that are dominated by mouth bar construction.

We suggest that sediment properties set delta front rugosity because changes in sediment properties shift the time scales of two fundamental processes that construct the delta: (1) distributary channel avulsion and (2) river mouth bar construction. For a given river mouth, if the time scale for river mouth bar construction is shorter than the time scale for channel avulsion, the channel will remain in a location long enough to construct a mature mouth bar, thus increasing delta front rugosity. Alternatively, if the channel avulsion time scale is shorter, the channels will relocate before constructing a mouth bar, decreasing delta front rugosity due to mouth bar growth. If both mouth bar and avulsion time scales are long, then the channels will simply prograde, creating a smooth delta front.

To test this idea, we quantified a theoretical river mouth bar building time scale (Trmb) to compare it against inline image. We modified the equation used by Jerolmack and Swenson [2007] and calculated Trmb as the ratio of the mouth bar volume (0.6b2h) to the depositional sediment flux ( βQt,ch):

display math(11)

where Qt,ch is the total sediment flux exiting a channel mouth. We adjust the mouth bar volume calculation to reflect previous findings that mouth bars are stable when they aggrade to a height equal to ~0.6h [Edmonds and Slingerland, 2007]. Not all of the sediment exiting the channel mouth contributes to mouth bar building; fine grains with slow ws and long advection lengths are transported far into the basin and contribute to the bottomset. To account for this process, we define a bypass fraction β that ranges from 0 to 1. It is determined by the fraction of grains with advection lengths smaller than or equal to the distance from the channel mouth to the mouth bar, which we assume to be equal to 5b, consistent with distances in this study. Once a delta reaches dynamic equilibrium (Figure 7), we calculate Trmb for every individual channel mouth and compared it to the delta's inline image.

The ratio Trmb/inline image characterizes the dominant behavior for a given channel mouth. We calculate this ratio for all channels on a given delta to characterize whether mouth bar growth or channel avulsion is the dominant process occurring at the channel mouth and thus constructing the delta. The percentage of channels with Trmb/inline image < 1 (prone to mouth bar building) peaks at intermediate values of D84 (Figure 10c). Considering this, we can explain the variability in delta front rugosity as a function of relative avulsion and mouth bar building time scales.

Deltas with the largest percentage of channel mouths with Trmb/inline image < 1 (intermediate-grained deltas with D84 = 0.04–0.4 mm) (Figure 10c) are dominated by the process of river mouth bar construction. This occurs in part because the relatively shallow gradient (Figure 8a) and intermediate percentages of cohesive sediment keep inline image long compared to Trmb. Furthermore, the grains are coarse enough with fast enough ws to encourage sediment deposition near the channel mouth, which increases β, and the number of channels is small so flow splits infrequently, which increases Qt,ch. Together, these effects decrease Trmb. Because the delta has many channels that remain in one location long enough for river mouth bars to grow, increased mouth bar growth and progradation create local delta front perturbations, leading to the most rugose delta fronts observed (Figure 10d).

Coarser-grained deltas (D84 = 0.4–4 mm) have fewer channel mouths with Trmb/inline image < 1 (Figure 10c) and are dominated by the process of distributary channel avulsion. Because inline image is generally shorter than Trmb, channels often relocate before a river mouth bar forms. Mouth bar building is further suppressed because the large number of channels (Figure 8b) decreases Qt,ch, thus increasing Trmb (equation (11)). Coarse-grained deltas have inline image values much shorter than the total modeled delta lifetime (T), and thus, many avulsions occur during delta growth. This leads to relatively even sediment deposition along the delta shoreline, resulting in smoother delta fronts relative to those created by the intermediate-grained deltas (Figure 10d).

Although fine-grained deltas (D84 = 0.01–0.04 mm) have fewer channel mouths with Trmb/inline image < 1 (Figure 10c) which suggests they are avulsion dominated, their inline image and Trmb values are both longer than the total delta lifetime (T). Thus, neither process occurs frequently during delta growth. inline image values are long because these deltas have shallow gradients (Figures 8a and 8e) and increased percentages of fine, cohesive sediment with high τce values that stabilize channels (Figure 10a). Trmb values are long because ws values are low and thus β values are high, and much of the sediment exiting the channel mouth bypasses mouth bar deposition. Instead, a few, stable channels confine the majority of water and sediment discharge to deep channel mouths, which promotes basinward progradation of a few, elongate channels [Kim et al., 2009; Edmonds and Slingerland, 2010; Falcini and Jerolmack, 2010; Rowland et al., 2010, Mariotti et al., 2013], creating deltas with smooth delta fronts (Figure 10d).

5.2.2 How Do Sediment Properties Control Delta Shape?

In our runs, finer-grained deltas have planform shapes that are more elongate, whereas coarser-grained deltas are more semicircular in shape (Figure 8d). We suggest that the fine-grained deltas are elongate in shape because they are created by a few channels prone to elongation via subaqueous levee progradation.

To test this idea, we calculate the percentage of channel mouths on each delta that have an unstable turbulent jet. Previous work suggests that when sediment-laden turbulent jets exiting channel mouths are unstable, they meander and increase their lateral diffusivity, leading to enhanced levee growth at the expense of mouth bar growth [Rowland et al., 2010; Falcini and Jerolmack, 2010; Mariotti et al, 2013; Canestrelli et al., 2014]. We follow the instability criterion used by Canestrelli et al. [2014], which suggests jet instability is a function of both the “river mouth” Reynolds number [Dracos et al., 1992] and the stability number [Van Prooijen and Uijttewaal, 2002; Socolofsky and Jirka, 2004; Rowland et al., 2009]. Following this criterion, a channel mouth's jet is unstable if

display math(12)

where C is the Chézy value used for hydrodynamic roughness (C = 45 m1/2 s−1 in our runs) and uch is the cross-section-averaged velocity at the channel mouth. Thus, turbulent jet stability is determined by both the channel mouth width to depth ratio and the velocity. While channel mouth velocity values vary minimally among the runs in this study (data not shown here), channel mouth width to depth ratios vary by 2 orders of magnitude (empty circles, Figure 10e), accounting for the majority of observed variation in jet instability among runs.

As D84 increases, there is a decrease in the percentage of channel mouths with an unstable turbulent jet (R2 = 0.70) (black circles, Figure 10e). Interestingly, an increase in percentage of channel mouths with unstable turbulent jets relates to a decrease in delta shape (A) (R2 = 0.47) (Figure 10f). This relationship shows that fine-grained, elongate deltas (Figure 8d) are formed by a larger percentage of channel mouths with unstable turbulent jets. The turbulent jets are unstable because of decreased channel mouth width to depth ratios (Figure 10e), which results from the increased percentage of fine grains in the sediment load, consistent with field observations [Church, 1992]. The high percentage of unstable jets promotes levee progradation and this, coupled with the observation of only a few channels in fine-grained deltas (Figure 8b), creates an elongate delta shape. Intermediate-grained deltas have fewer channel mouths with unstable turbulent jets (Figure 10e), which promotes deposition nearshore and formation of river mouth bars rather than prograding levees. Deltas dominated by mouth bar growth bifurcate frequently, creating semicircular shapes (Figure 10f).

5.3 Process-Based Model for the Effect of Sediment Properties on Delta Morphology

We propose that the observed morphological transition related to the dominant grain size and percent cohesive sediment occurs because variations in these sediment properties set the number of channel mouths and the dominant process operating on the delta.

Coarse-grained deltas in this study have steep topset gradients, many channel mouths, relatively smooth delta fronts, and semicircular delta shapes. The coarse-grained, noncohesive sediment inputs construct deltas with steep topset gradients (Figure 8a) in order to create higher bed shear stresses to transport the coarser grains with higher τce values. At dynamic equilibrium (Figure 7a), these steep deltas aggrade faster per unit progradation, which leads to increased channel avulsion frequency (shorter inline image values) (Figure 10a) and more channel mouths (Figure 10b). Although the channel mouths have relatively large width to depth ratios (Figure 10e) that promote river mouth bar growth over levee growth [e.g., Canestrelli et al., 2014], and the grains have fast ws values that promote nearshore deposition, the large number of channels splits the flow and decreases sediment discharge exiting each channel mouth, lengthening Trmb. The result is a delta where the majority of channel mouths have Trmb/inline image > 1 (Figure 10c) and are thus dominated by avulsion. The combination of many channels and frequent avulsions creates a delta with a relatively smooth delta front (Figures 8c and 10d) and a semicircular shape (Figures 8d and 10f).

Intermediate-grained deltas in this study produce intermediate topset gradients and numbers of channel mouths, rugose delta fronts, and semicircular delta shapes. The intermediate-grained sediment inputs have smaller τce values and intermediate ws values, creating deltas with shallower topset gradients (Figure 8a) than those of coarser-grained deltas, and thus channels with longer inline image values (Figures 10a). Intermediate numbers of channel mouths keep sediment flux per channel mouth high, and intermediate ws encourages sediment deposition in the mouth bar zone. This creates a higher percentage of channels with Trmb/inline image < 1 (Figure 10c). The dominance of mouth bar growth creates a bifurcated channel network with an intermediate number of channel mouths (Figure 8b) that construct mouth bars along the entire delta shoreline, creating a rugose delta front (Figures 8c and 10d) and a semicircular shape (Figures 8d and 10f).

Fine-grained sediment inputs create deltas with shallow topset gradients, a few channel mouths, smooth delta fronts, and elongate delta shapes. The fine-grained, cohesive sediment inputs have grain sizes with high τce values and slow ws values. The slow ws leads to long advection lengths of the fine grains, which are transported far into the basin, dispersing the sediment and creating shallow topset gradients (Figure 8a). The increased τce values due to the cohesive sediment stabilizes channels, increasing inline image values (Figure 10a) and decreasing the number of channel mouths (Figure 10b). The channels have small width to depth ratios (Figure 10e) which leads to creation of unstable turbulent jets exiting most channel mouths (Figure 10e). The result is a delta with a few channel mouths (Figures 8b and 10b) dominated by the process of levee progradation, creating a smooth delta front (Figures 8c and 10d) and elongate delta shape (Figures 8d and 10f).

Here we have shown that varying the character of the sediment properties shifts the balance of processes constructing a delta and thus sets the channel network and delta planform morphology. We do not mean to imply, for example, that all intermediate-grained deltas are dominated by mouth bar construction. After all, additional variables exert control on many of the relationships that form the process-based model presented here. For example, an increase in Q/Qt has been shown to increase transport slopes [Parker et al., 1998]. Thus, an intermediate-grained delta may be avulsion dominated if its topset gradient was increased by significantly lowering Q relative to Qt.

The process-based model presented here provides a previously lacking mechanistic understanding of the effects of sediment properties on delta channel network and planform morphology that is in agreement with qualitative observations linking grain size to delta morphologies [McPherson et al., 1987; Orton and Reading, 1993]. For example, it was previously shown that an increase in grain size from silt to gravel results in a morphological transition from elongate to semicircular delta shapes [McPherson et al., 1987; Orton and Reading, 1993]. Here we show that this transition is related to an increase in topset gradient and subsequent increase in channel avulsion frequency relative to the frequency of bifurcations around mouth bars, leading to a large number of channels on the delta topset that deliver sediment evenly across the shoreline, creating a semicircular shape.

6 Implications

If sediment properties are as important as this study indicates, then a natural question to ask is what controls the properties of the incoming sediment load. To a first order, variations in properties of the sediment input entering a delta apex result from differences in upstream catchment characteristics. These include source properties that build the initial grain-size distribution (e.g., lithology and degree of weathering) and modification of the distribution downstream by fluvial processes through abrasion and selective deposition [e.g., Ferguson et al., 1996; Rice, 1999; Fedele and Paola, 2007]. In general, larger catchments with low relief deliver more predominantly fine sediment, whereas smaller catchments with high ruggedness deliver more coarse-grained sediment [Milliman and Syvitski, 1992; Orton and Reading, 1993; Mulder and Syvitski, 1995].

Our results suggest that to a first order, active margin coasts that are generally dominated by steep, high relief catchments with short transport distances should be dominated by coarse-grained deltas that are semicircular with many active channels. Passive margin coasts with large drainage basins that drain low-relief continental interiors and have long transport distances should deliver finer grains to the coasts and produce elongate deltas. With these relationships in mind, the model presented here may be used to connect delta morphology to catchment area, though additional work developing such predictive capabilities is necessary.

7 Conclusions

We show that variations in the dominant grain size and percent cohesive sediment of the incoming sediment load exhibit a significant control on deltaic processes and morphology. Analysis of 36 numerically modeled deltas shows that deltas undergo a morphological transition as grain size increases from elongate planforms with shallow gradients and a few stable channels to semicircular planforms with steep gradients and many mobile channels. This morphological transition occurs because the character of the sediment properties determines the number of channel mouths and the dominant process that occurs on the delta and sets the morphology.

Coarse-grained, noncohesive sediment inputs create steep topset gradients due to high critical shear stresses for erosion. Steeper gradients lead to shorter channel time scales (inline image) and a large number of channel mouths. The large number of channel mouths splits flow frequently, leading to decreased sediment discharge per channel mouth (Qt,ch) and increased mouth bar growth time scales (Trmb). The result is a delta with many channel mouths that are dominated by the process of channel avulsion (Trmb/inline image >1), which creates a semicircular delta and relatively smooth delta front.

Intermediate-grained sediment inputs have intermediate topset gradients, inline image values, and numbers of channel mouths. Less flow splitting within the channel network creates increased Qt,ch and thus decreased Trmb values relative to coarser-grained deltas. The lower critical shear stresses for erosion relative to finer-grained sediment inputs creates channels with larger width to depth ratios which are more likely to create river mouth bar deposits than levee deposits. The result is a delta with many channel mouths that are dominated by the process of river mouth bar growth (Trmb/inline image <1), which creates a semicircular delta and rugose delta front.

Fine-grained, cohesive sediment inputs create shallow topset gradients due to slow settling velocities. High critical shear stresses for erosion stabilize channel banks, leading to longer inline image values, smaller width to depth ratios, and only a few channel mouths. The decreased width to depth ratios create unstable turbulent jets exiting channel mouths. The result is a delta with few channel mouths, the majority of which are dominated by the process of levee growth, which creates an elongate delta and smooth delta front.

Notation
A

delta shape, nondimensional.

b

channel width, m.

B

delta width, m.

ci

mass concentration of the ith sediment fraction, kg m−3.

C

Chézy value, m1/2 s−1.

D50

median grain size, mm.

D84

representative dominant grain size, mm.

Di

grain size of the ith sediment fraction, m (unless otherwise noted).

fi

weight percent of the ith sediment fraction, nondimensional.

Fd,i

depositional flux of the ith cohesive sediment fraction, kg m−2 s−1.

Fe,i

erosive flux of the ith cohesive sediment fraction, kg m−2 s−1.

g

acceleration due to gravity, m s−2.

Gch

ratio of channelized grid cells still active after some time tch, nondimensional.

h

channel depth, m.

inline image

average channel depth, m.

L

delta length, m.

Pm:Pr

proxy for ratio of marine power to river power, nondimensional.

Q

water discharge, m3 s−1.

q

water discharge per unit width, m2 s−1.

qb,i

bedload discharge per unit width of the ith sediment fraction, m2 s−1.

Qt

total sediment discharge, m3 s−1.

inline image

time-averaged total sediment discharge, m3 s−1.

Qt,ch

total sediment discharge exiting channel mouth, m3 s−1.

R

submerged specific gravity of sediment, ρs/ρw − 1, nondimensional.

Sk

skewness of the grain-size distribution, nondimensional.

t

current time in delta growth, year.

tch

time since channel initiation, year.

T

total modeled delta lifetime, year.

TA

predicted channel avulsion time scale, year.

inline image

measured average channel time scale, year.

Trmb

theoretical river mouth bar formation time scale, year.

u

depth-averaged velocity, m s−1.

uc,i

critical depth-averaged velocity of the ith sediment fraction; m s−1.

uch

cross-section-averaged velocity at channel mouth, m s−1.

ws

settling velocity, m s−1.

β

mouth bar sediment bypass fraction, nondimensional.

εs,x,i, εs,y,i, and εs,z,i

eddy diffusivities in the x, y, and z directions, respectively, of the ith sediment fraction, m2 s−1.

inline image

channel aggradation rate, m yr−1.

λ

decay constant, inverted for mean lifetime of an “active channelized grid cell,” year.

ν

kinematic viscosity coefficient of water, m2 s−1.

ρs

specific density of sediment, kg m−3.

ρw

specific density of water, kg m−3

σ

standard deviation of the grain-size distribution, φ.

τ0

bed shear stress, N m−2.

τce

critical shear stress for erosion, N m−2.

τce(C)

critical shear stress for erosion of cohesive sediment fractions, N m−2.

τcd(C)

critical shear stress for deposition of cohesive sediment fractions, N m−2.

φ

grain size phi value, nondimensional.

ψ

superelevation of a channel relative to h for an avulsion to occur, nondimensional.

Acknowledgments

The data for this paper are available upon request from the authors. We would like to acknowledge funding from National Science Foundation grant OCE-1061380 and grant FESD/EAR-1135427 awarded to D.A. Edmonds. We also thank Rudy Slingerland, Alex Burpee, Jim Best, Dan Parsons, and Liz Hajek for valuable comments. We thank the associate editor, Wonsuck Kim, and reviewers Andy Wickert, Wout van Dijk, and two anonymous reviewers for comments that greatly improved this manuscript.