Controls on river delta formation; insights from numerical modelling

Earth and Planetary Science Letters 302 (2011) 217–226

Contents lists available at ScienceDirect

Earth and Planetary Science Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e p s l

Controls on river delta formation; insights from numerical modelling Nathanaël Geleynse a,⁎, Joep E.A. Storms a, Dirk-Jan R. Walstra b,c, H.R. Albert Jagers c, Zheng B. Wang b,c, Marcel J.F. Stive b a b c

Department of Geotechnology, Delft University of Technology, 1, Stevinweg, 2628 CN, P.O. Box 5048, Delft, The Netherlands Department of Hydraulic Engineering, Delft University of Technology, 1, Stevinweg, 2628 CN, P.O. Box 5048, Delft, The Netherlands Deltares (former Delft Hydraulics Laboratory), 185, Rotterdamseweg, 2600 MH, P.O. Box 177, Delft, The Netherlands

a r t i c l e

i n f o

Article history: Received 28 August 2010 Received in revised form 1 December 2010 Accepted 6 December 2010 Available online 31 December 2010 Editor: P. DeMenocal Keywords: river deltaic networks coastal waves sedimentary stratigraphy

a b s t r a c t Primary hydraulic and sedimentary controls on river deltas were previously postulated in classification schemes in the field of sedimentary geology, based on many field observations. However, detailed mechanistic models were restricted to the river-dominated delta class. Analogous to the concept of morphological conditioning, here we show that antecedent stratigraphy controls morphometrics of prograding river-dominated delta distributary networks under steady sea level. Further, we use coupled hydrodynamic-morphodynamic-stratigraphic detailed numerical modelling to assess the influence of windgenerated waves and tides on clastic river delta formation. Our synthetic simulations show that deltas forming under mere riverine forcing prograde via sequences of mouth-bar induced flow bifurcation and upstream channel shifting. Windwave action suppresses sequestration of fine sediments on the developing delta plain, entailing relatively smooth shorelines, perturbed by a limited number of distributary channels. In contrast, tide-influenced river deltas are found to prograde mainly via lengthening of initially-formed, relatively stable distributaries, as well as being characterized by cyclicity in deposits (interbedding of sands and silts). These results provide a framework for physics-based river delta modelling under various environmental conditions. Particularly, our findings suggest that relatively low-energetic basin conditions can already significantly impact morphological and stratal patterns of prograding river deltas. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Presently, river deltas are receiving much attention because of their multi-faceted significance to mankind. In particular, their response to coupled atmospheric forcing and relative sea level change is aimed to be understood (Syvitski et al., 2009). However, future dynamics of deltas resulting from natural or human activity can only be challenged if their present state can be understood. Likewise, reconstructions of past behaviour of deltas naturally rely on what is known of present-day river deltas. From field observations, it is generally inferred that the morphology of deltas is governed by hydraulic forcings. The relative importance of a river, coastal windwaves and tidal waves to a delta's planform was embodied by a diagram that, throughout the recent past, has become a classic delta classification scheme (Galloway, 1975). Later, this tripartite scheme was extended to include sediment calibre as a fourth principal axis (Orton and Reading, 1993) (Fig. 1a). Consequently, a quasi-fourth spatial dimension arises in traditional three-dimensional Cartesian landform models in that the sediment subsurface requires schema-

⁎ Corresponding author. Tel.: +31 15 2784419; fax: +31 15 2783328. E-mail address: [email protected] (N. Geleynse). 0012-821X/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.epsl.2010.12.013

tization, an issue that presently has not been routinely addressed (Hutton and Syvitski, 2008). Recently, the ability of physics-based numerical morphodynamicstratigraphic modelling to address formation of the river-dominated delta class has been demonstrated (Geleynse et al., 2010). Moreover, the importance of sediment cohesiveness to river delta morphology has been shown (Edmonds and Slingerland, 2010). It is the sediment in downstream drainage basins that mainly reflects characteristics of upstream drainage basins, these being linked by the fluvial system (Rodríguez-Iturbe and Rinaldo, 2001; Schumm, 1977). Accordingly, it follows that a deposit can only be wholly understood if its feeder system is known. The role of this feeder system was already exemplified in a review of a plethora of delta classification schemes (Nemec, 1990). However, it was only recently that we found upstream controls (downstream-directed river bar and meander migration) on simulated three-dimensional river delta dynamics (Geleynse et al., 2010). Also, water discharge variability at a delta's head was shown to control the number of distributary channels on a delta plain (Edmonds et al., 2010). Further, the stability of river channel bifurcations was assessed to be partially set by upstream controls (river channel width-to-depth ratio and planform curvature) (Kleinhans et al., 2008). Hence, it can readily be understood that a delta does not merely ensue from a point-source, a fixed efflux at some point in time. Instead, it forms an integral part of a dynamic

218

N. Geleynse et al. / Earth and Planetary Science Letters 302 (2011) 217–226

Fig. 1. A schematized standard classification scheme wherein deltas are grouped according to their controlling hydraulic forcing type (river, tides, windwaves) and sedimentary forcing type (sand, silt, mixture, in flux and in subsurface) (Orton and Reading, 1993). In principle, the present study focuses on the upper part of this triangle (a), though pinpointing the position of a delta in this spectrum is not without difficulty, because of unsteadiness of these hydraulic forcings and of other (human) factors known to govern antecedent and present-day delta morphology (Syvitski and Saito, 2007). This observational scheme can be naturally represented by a physics-based numerical delta scheme, starting from highly-schematized conditions (b).

larger-scale terrestrial-oceanic system (Swenson et al., 2005), the attributes of which (e.g., a delta) may deform through a wide range of processes, acting across a wide range of frequencies and lasting for a wide range of periods, including high-energy, low-frequency, daily events (Hutton and Syvitski, 2008; Paola et al., 1992; Storms, 2003; Swenson et al., 2005). Here, we analyze simulations from a numerical delta scheme, based on conservation principles, aiming for identification of key factors controlling deltas. We elucidate primary ingredients of abovementioned delta schemes, based on many field observations, namely: (1) the role of subsurface sediment composition, and (2) coupling of feeder channel dynamics to alluvial deltas, formed by (3) riverine sediments, and while being deformed by windwave or tidal wave action. 2. Methods To investigate the formation of clastic river deltas under different hydraulic and sedimentary forcing mechanisms, we use highresolution hydrodynamic-morphodynamic-stratigraphic numerical modelling based on a continuous ansatz. A two-dimensional hydrodynamic model is coupled to a three-dimensional morphodynamicstratigraphic model within the Delft3D numerical modelling environment (Lesser et al., 2004). 2.1. Modelling of river flow, coastal windwaves and tides Delta hydrodynamics are represented by the unsteady nonlinear depth-averaged shallow-water equations, a simplified form of the well-known Navier-Stokes equations for incompressible free-surface flow. For the cases of windwaves modifying river outflow, these conservation equations are expressed in Generalized Langrangian Mean coordinates (Andrews and McIntyre, 1978). The depth-averaged wave-induced forcing term is approximated through a dissipation term formulation (Dingemans et al., 1987). The SWAN wave model (Booij et al., 1999) is used to simulate the propagation and dissipation of organized wave energy. The changing wave spectrum is represented by the spectral action density balance formulation. Herein, the source term of energy density contains the dissipation of wave energy only; wave growth due to wind and computationally-demanding nonlinear (quadruplet and triad) wavewave interactions are neglected. Dissipation due to bed friction,

breaking and whitecapping are modelled through an empirical friction model (Hasselmann et al., 1973), a bore model (Battjes and Janssen, 1978) and a pulse-based model (Hasselmann, 1974), respectively. For all cases presented herein, a relatively low-energy wave climate is imposed at the open basin boundaries. The offshore wave field is oriented perpendicular to the undisturbed shoreline and is characterized by a significant height of 1.0 m and a peak period of 5.0 s. To model the effect of secondary flow on the depth-averaged flow, additional shear stresses in the shallow-water equations are modelled according to the approach of Kalkwijk and Booij (1986). Adopting Boussinesq's eddy viscosity concept, the horizontal Reynolds stresses are related to gradients of the depth-averaged velocities. To determine the horizontal turbulent diffusivity of momentum, several turbulence models of increasing complexity exist (e.g., Rodi, 1993), however, for simplicity, herein we assume it to be constant (1.0 m2s− 1). Wind shear stress at the surface and Coriolis force are neglected. Further, note that possible intake of water by, for example, precipitation, tributary inflow or evaporation, is not considered. At closed boundaries, river banks, tide-induced drying of bars and delta plain, or elements below a critical flow depth (0.1 m), a free-slip condition is assumed. Bed shear stresses (enhanced for waveconditions) are related to components of the depth-averaged velocity vector by Chézy's relation. Herein, the Chézy-coefficient is defined to vary with local flow depth and space-invariant Manning's n (0.03 sm− 1/3). Generally, the latter parameter accounts for lumped effects, such as bed slope, bed forms, parent material, vegetation and large-scale channel roughness (Chow, 1959), however, for simplicity, its spatio-temporal variability due to, for example, vegetation dynamics is neglected herein. For the tidal simulations (resembling the semi-diurnal lunar tide), the amplitude of the tidal water level gradient and phases are prescribed at the lateral basin boundaries, in conjunction with the tidal amplitude (1.5 m) and a linearly-varying phase at the offshore boundary, in order to avoid spurious flow fields (Roelvink and Walstra, 2004). At the upstream river boundary, the flow condition is given by a time-invariant water discharge (2000 m3s− 1). This total water discharge (Q) is then distributed over the N boundary cross-stream cells according to a weighting function, accounting for differences in N

3=2

local flow depth (hi): Qi = h3/2 i Q / ∑ hj

.

j=1

2.2. Numerical parameters and initial geometry The shallow-water equations are discretized on a staggered uniform finite-difference grid and solved by means of an Alternate Direction Implicit time integration method (Stelling and Leendertse, 1991). A constant time step of 15 s is chosen, for reasons of accuracy and stability. Here, the same Cartesian grid cells (50 by 50 m) are used to discretize the flow- and wave-field, however, to minimize boundary distortions, the geographical wave domain is extended at the open basin boundaries. The computational spectral grid is specified by the frequency domain (here defined between 0.05 and 1.0 Hz with 24 logarithmically-distributed bins) and the directional domain (here defined as the full 360° with a spectral resolution of 10°). Currents, waves and bed levels are simulated at the same grid; hence no interpolation inaccuracies are introduced. Flow and sediment transport fields, current-wave interaction and bed level updating are executed at each time step, while the wave field is updated over a multiple of time step (Roelvink, 2006). The spatial domain features an initially straight river of length (1200 m), width (550 m) and depth (2.5 m), embanked by its erodible floodplain, and debouching into a linearly-sloping basin (8000 m in cross-shore direction and 10,000 m in long-shore direction). See Fig. 1b for a schematization of the model set-up.

N. Geleynse et al. / Earth and Planetary Science Letters 302 (2011) 217–226

219

2.3. Sediment characteristics and transport

2.5. Hydraulic and sedimentary boundary conditions

Two sediment fractions are modelled. Herein, these are defined by median diameter, fall velocity, specific density, dry bed density and critical shear stress for erosion. A fine sediment fraction (specific density = 2650 kgm− 3, dry bed density = 500 kgm− 3), which potentially exhibits a certain shear strength, and a coarser sediment fraction (specific density = 2650 kgm− 3, dry bed density = 1600 kgm− 3), which in principle is non-cohesive, are modelled. Based on particle size only, these fractions represent coarse silt (50 μm) and sand (200 μm), respectively. However, erodibility of natural sediment beds is not only governed by size and weight of solid particles, but also by network structure and cohesion. This particularly holds for fine sediment beds (clay and silt) (van Ledden et al., 2004). The present model assumes no particular bed network structure, that is, entrainment of both sediment fractions is treated independently, unless preferential uptake leads to depletion of that fraction in the transport layer (supply-limited condition), the upper sediment layer through which exchange of sediments with the water column occurs. Sand transport is modelled according to the algebraic formulation of Engelund and Hansen (1967), whereby effects of longitudinal and transverse bed slopes are accounted for, following the Bagnold-Ikeda expressions (Garcia, 2006; van Rijn, 1993). Silt transport is computed through a depth-averaged advection-diffusion formula. The erosion and deposition rates follow from the well-known Krone-Mehta-Partheniades formulations (Winterwerp and van Kesteren, 2004), wherein the fall velocity, critical shear stress for erosion, the erodibility coefficient and the probability of deposited silt to adhere to the bed are set to 0.0015 ms− 1, 0.5 Nm− 2, 0.0001 kgm− 2 s− 1 and 1, respectively. The transport equation is discretized with a finite volume approximation. Sediment transports by undertow and wave asymmetry, as well as aeolian transport are neglected here. Accordingly, barrier formation (Bhattacharya and Giosan, 2003; Stutz and Pilkey, 2002) may appear impossible to model here a priori, however, its generating mechanism is hitherto unresolved (van Maren, 2005, and references therein).

Autogenic dynamics of natural systems have been increasingly targeted as a potential explanation for observed landform diversity and self-organized behaviour (Kim et al., 2006; Paola et al., 2009), notwithstanding the salient complexity of demarcation of a finite piece of the real world (Thorn and Welford, 1994), be it a single landform or a series of connected landforms; a landscape (Allen, 2005; Burrough and McDonnell, 1998). Herein, we continue along these lines by imposing time-invariant hydraulic boundary forcing to our mathematical system, expressing physical reality in approximate form. To ensure a most straightforward comparison of simulated deltas under different forcing types, all shared model parameters are assigned identical values (see 2.1–2.4). To assess the influence of different subsurface sediment composition to modelled river and delta morphology, while the initial total thickness of the sediment layer and the relative volumetric fraction across and along the vertical are held constant, the sand-silt volume ratio is varied to address three scenarios: an initial (1) sand-dominated (100% sand, 0% silt), (2) mixed-load (50% sand, 50% silt) and (3) silt-dominated (25% sand, 75% silt) sedimentary system. At the upstream boundary, steady water and silt fluxes (0.04 kgm− 3) are imposed. The sand influx is given by a Neumann condition, a zero-condition on the derivative of the flux at the inflow boundary, such that channel bed levels across the boundary section are steady. Accordingly, for all cases a welladapted flow enters the spatial domain. At the downstream basin boundary, three forcing scenarios are highlighted: a time-invariant (1) still water level, (2) tidal wave and (3) wind-generated short wave field.

2.4. Subsurface sediment composition The sediment subsurface is schematized according to Ribberink's multi-layer concept (Garcia, 2006; Ribberink et al., 2002). The transport layer has a constant thickness (0.2 m), while a maximum number of 75 layers (each 0.1 m thick, except a layer of variable thickness (≤0.1 m), just below the transport layer) tracks sedimentary composition along the vertical through time. Sediment fluxes are reduced if the immobile substrate below the sediment package is reached. In view of observed significant scouring occurring in river reaches and delta channels (Best and Ashworth, 1997; Eilertsen and Hansen, 2008), to allow for fully alluvial conditions (cf. Edmonds and Slingerland, 2010; Geleynse et al., 2010), the entire spatial domain is initially made up of 20 m of sediment (Fig. 1b), accounting for differences in sand-silt mixtures due to differences in porosity. The quantity of sand and silt available at the transport layer is updated for each half ‘hydrodynamic’ time step via bookkeeping for the control volume of each computational cell. Bed level changes are linearly upscaled (60), to decrease computational time, assuming that timescales of relaxation of the bed are well beyond those for the flow field. A numerical substitute for erosion of inactive, dry bank and bar cells is used; sediment (per fraction) which is eroded from an active, wet grid cell is (uniformly) replenished from its neighboring inactive cells, which, thence may become active. If a certain sediment fraction is not (sufficiently) present in these adjacent cells, the eroded volume is replenished with a volume that is made up (enriched with) the other sediment fraction.

Fig. 2. Sand and silt fluxes, integrated over the river-width, at the initial river efflux position, versus simulated time (2-months resolution, total simulated time T = 7.5 yr), and grouped per (coupled) hydraulic forcing type; river (a), river-tides (b), riverwindwaves (c). The ratio of sand and silt load transport rate changes for different initial schematizations of the river bed (line colors correspond to initial vol. % sand) and with time (hinting at morphodynamics of the river channel and, herewith associated, sand inflow at the upstream boundary). Here, for a tide-modulated river outflow, (suspended) silt fluxes prevail, irrespective of initial river subsurface sediment composition and time. Symbols (R1-R3,T1-T3,W1-W3) refer to cases presented in Fig. 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

220

N. Geleynse et al. / Earth and Planetary Science Letters 302 (2011) 217–226

3. Results and discussion 3.1. Coupled hydraulic-sedimentary forcing to River-Deltas The modelled morphology of river-deltas for different combinations of the above-listed hydraulic and sedimentary forcing types are compared at the time when sediment fluxes at the initial river efflux position attain approximate steady-state (Fig. 2). From our model experiments, the differential effects of tides and windwaves to the morphology of a prograding river delta can clearly be assessed (compare deltas along the vertical axis of Fig. 3). Additionally, we

underline that sedimentary forcing adds another dimension (cf. Fig. 1a) that needs to be taken into account while explaining differences in morphodynamics of river delta distributary networks under different hydraulic forcing (compare deltas along the horizontal axis of Fig. 3). 3.2. Effects of wind-generated waves Generally, normally-incident wave fields lead to smooth and virtual axisymmetric delta fronts (Fig. 3, upper panel), in agreement with field observations and associated modelling (Jerolmack and

Fig. 3. Simulated delta morphology for different combinations of hydraulic and sedimentary forcing types after 7.5 simulated years. Note that only the green colors depict emerged features (N0 m mean water level (MWL)), which would generally be registrated by air, or space-borne imagery. Solid gray, black and white lines refer to stratigraphy sections in Fig. 4a, b, c, respectively. ‘Sand’, ‘Sand-Silt’ and ‘Silt’ refer to 100%, 50% and 25% sand (by volume) in the initial subsurface, respectively. Greek symbols represent: mean distributary depth (α) and sinuosity (β), delta front (shoreline) roughness (γ), subaerial delta volume (δ), river valley width () and longitudinal slope (ζ). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

N. Geleynse et al. / Earth and Planetary Science Letters 302 (2011) 217–226

221

Fig. 4. Simulated stratal response of deltas and computed water levels. (a) refers to stratal response after 2 simulated months, (b) and (c) after 7.5 simulated years. The dotted black lines in (b) and (c) depict the initial bed levels at − 3.5 m mean water level (MWL). Initially, a 20 m-layer of fully-mixed sediment is present in the entire spatial domain. Note the fine model resolution; 50 m in the horizontal directions and 0.1 m in the vertical direction. For symbols and locations of sections, refer to Fig. 3. Greek symbols represent: mean distributary depth (α), cross-delta mean water surface slope (η) and trapping efficiency of fines on the delta plain (θ). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Swenson, 2007), and results from other computational delta models (Ashton and Giosan, 2007; Seybold et al., 2007). Delta-front geometry is closely linked to dynamics of the delta-plain distributary network (Geleynse et al., 2010; Hoyal and Sheets, 2009; Kim and Jerolmack, 2008; Martin et al., 2009; Reitz et al., 2010), which in turn, is also governed by magnitude and composition of the total sediment flux at the river efflux, and by the subsurface sediment composition. Here, windwaves reduce subaerial delta progradation rates, consistent with Syvitski and Saito (2007) and Swenson (2005), by limiting sequestration of fines on the delta plain, hence steepening the delta-front, in accordance with field observations (Syvitski and Saito, 2007). Windwaves also seem to reduce the number of active delta-plain distributaries, though this cannot be ascertained here, due to the present neglect of wave-induced sediment transport. The latter assumption may be justified for considered low offshore wave heights (1 m), with onshore transport by asymmetry of shoaling waves (slightly) exceeding offshore transport by bed return flow under breaking waves (e.g., van Maren, 2005). In addition to deposition associated with large bifurcation angles (Wright, 1977), induced by bed friction and lateral mixing of momentum, in turn associated with shallowness of the initial efflux, windwaves enhance shore-parallel orientation of initial distributary channels (Fig. 5a) by deflecting plumes at distal ends of distributaries (Fig. 5b). Wave-current interaction results in local irregularities in wave height patterns, contrasting weaker riverine flow zones that display relatively smooth wave height contours (Fig. 5c). Here, with simulated time, steady riverine discharge keeps some channels open for routing of sediment to the prograding shoreline, however, these do not necessarily

bifurcate at their termini (Fig. 5a, Fig. 3, upper panel), an observation that supports the hypothesis that windwaves act to suppress mouthbar formation, and mouth-bar induced distributary formation (Bhattacharya and Giosan, 2003; Jerolmack and Swenson, 2007). Based on their extensive database, (Syvitski and Saito, 2007) found that a river's (maximum) discharge is the strongest predictor for the number of distributary channels on a delta plain. Accordingly, future investigations must pinpoint the role of unsteady and lower river discharge relative to wave power in determining characteristics of the distributary network structure, such as the number of distributaries formed by avulsion and flow bifurcation at the shoreline (Jerolmack, 2009; Jerolmack and Swenson, 2007; Swenson, 2005). Moreover, efforts are needed to address controls on the formation of deltaic barriers (e.g., spits), that are inferred not to be restricted to transgressive settings (Bhattacharya and Giosan, 2003). Examples include an assessment of the effects of (i) wave field characteristics (e.g., fraction of high-angle waves and directional asymmetry of wave fields; Ashton and Giosan (2007)), (ii) alongshore supply of sediment that is not directly derived from the active river (e.g., from abandoned delta lobes, erosional headlands or low stand shelf sands; Bhattacharya and Giosan (2003)) and (iii) relative contribution of cross-shore waverelated transport processes (e.g., with respect to cross-shore bed profile; van Maren (2005)), for different sediment fractions. 3.3. Stratigraphic conditioning Combined with sand fluxes towards the ensuing delta being nearly identical for the first two simulated months (Fig. 2), wave-induced

222

N. Geleynse et al. / Earth and Planetary Science Letters 302 (2011) 217–226

abandoned channels on an experimental fan surface were assessed to act as attractors for future flow (Reitz et al., 2010). Furthermore, based on observations from natural and experimental cohesive deltas, it has been suggested that deltas may be more erosive than currently thought (Best and Ashworth, 1997; Hoyal and Sheets, 2009; Shaw and Mohrig, 2009). The herein uniformly modelled, high-porosity, unconsolidated sediment (van Rijn, 1993) may depict significant spatio-temporal variability in real-world river-deltas, thereby controlling (succeeding) delta morphodynamics and stratal response. The presence of fines further governs delta morphometrics; mean distributary depth (α) and sinuosity (β) as well as delta front (shoreline) roughness (γ) increase for increasing subsurface's volumetric fraction of fines (Fig. 3, Fig. 4b), the latter and former being qualitatively similar to findings from laboratory experiments (Hoyal and Sheets, 2009; Martin et al., 2009). Note that the aboveobserved trend in sinuosity of distributaries is most apparent for deltas prograding under mere riverine forcing, since tide and windwaves reduce trapping efficiency of fines on the delta plain (θ) (Fig. 4c). Further, for increasing subsurface's volumetric fraction of fines, subaerial delta volume (δ) and cross-delta water surface slope (η) decrease (Fig. 3, Fig. 4b). 3.4. Distributive versus elongated channels

Fig. 5. Simulated initial developments of a wave-influenced river delta; (a) bed level (zb), (b) mean flow velocity field (b UN), thresholded at 0.1 m/s to highlight deflection of sediment plumes at the distal ends of some distributaries and (c) significant wave height field (Hs) after 14 simulated months, for a sediment subsurface that is initially made up of 50% sand. The upper boundary of the Hs plot corresponds to the general unperturbed shoreline (see bounding box in (b)). The two-sided white arrow indicates wave-current interaction, resulting in local constancy of wave height. Also, note the general absence of mouth bars at the distal ends of distributaries; when formed, these are typically reworked by wave action. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

offshore dispersal of fines, allow us to unambiguously constrain the role of subsurface sediment composition to delta morphology. Deltaplain distributaries are increasingly incisive for basements that are predominantly made up of fines (Fig. 4a). This finding appears consistent; for simulated river-dominated deltas (Fig. 3, lower panel), fines delivered at the river efflux are sequestered on the delta plain, thereby exhibiting sorting patterns that are in agreement with many observations from the Mississippi and Rhine-Meuse deltas (Gouw and Autin, 2002; Törnqvist et al., 1996), while distributaries forming in relatively fine sediments incise deeper, compared to their sandy counterparts (Fig. 4a). This basement incision, also corroborated by observations from the Wax Lake and Atchafalaya River deltas (Roberts et al., 1997; Shaw and Mohrig, 2009), points at the importance of antecedent basin stratigraphy to delta-plain morphology and distributary channel stability. Such stratigraphic preconditioning is analogous to the concept of morphological preconditioning (Lane and Richards, 1997), as also recently identified from modelling, in that

3.4.1. Delta planform controlled by initial stratigraphy An outstanding question regarding the formation of deltaic channels, in addition, and closely tied to their afore-outlined potential incisive mode (3.3), is whether initiation of these channels is characterized by a preferential deposition direction. Preferential deposition along a prograding channel axis implies a lower bound to its distributive character that has traditionally been suggested to be typical for deltas. This ‘elongate’ delta progradation type (Galloway, 1975) has been challenged by analogy with a leaky pipeline, to itemize conditions most likely needed to examine the elongated channel that feeds the modern Mississippi delta, as well as some lacustrine deltas (Kim et al., 2009). In a recent contribution, dimensionless excess shear
stress [τE = τ0 −τceðcÞ = τ0 , where τ0 is the spatio-temporal mean bed shear stress at the river efflux and τce(c) is the critical shear stress for erosion of the cohesive sediment fraction] and relative cohesive sediment influx [Qrc = Qc = Qn , where Qc and Qn is the mean cohesive and non-cohesive sediment flux at the river efflux, respectively] were introduced to control this preferential deposition direction and channel sinuosity (Edmonds and Slingerland, 2010). For relatively strongly cohesive sediment settings (relatively high τce(c), with respect to τ0 and relatively high Qrc), deltas were found to exhibit a more elongate and sinuous planform. Contrarily, deposition across the channel axii under high τE and relatively low Qrc was suggested to give rise to fan-like deltas (Edmonds and Slingerland, 2010), also termed ‘space-filling deltas’ (Kim et al., 2009). Apart from the notion that deltas that plot in the lower extreme of this parameter spectrum may exhibit relatively large planform variability for relatively small changes in τE or Qrc, our model river-dominated deltas reveal, that for identical critical erosional shear stress of the fine sediment fraction (τce(c) = 0.5 Nm− 2), and relatively small differences in τ0 (2.57 ± 0.62 Nm− 2) and Qrc (1.90 ± 1.54), all being at the lower end of reported ranges (τce(c) = [0.10–2.60] Nm− 2, τ0 = [4.22–10.68] Nm− 2 and Qrc = [0.12–24.78], respectively) (Edmonds and Slingerland, 2010), basin subsurface sediment composition provides a third dimension for controlling river delta morphometrics. 3.4.2. Delta planform controlled by initial efflux geometry The river-dominated delta simulations presented above belong to the ‘space-filling’ delta class (Fig. 3, lower panel), that is, are predominantly governed by channel formation through (i) mouth-bar induced flow bifurcation at the prograding shoreline (Edmonds and Slingerland, 2007; Wright, 1977) and (ii) avulsion (reoccupation of an

N. Geleynse et al. / Earth and Planetary Science Letters 302 (2011) 217–226

abandoned channel or dissection of a levee or mouth-bar) (Edmonds et al., 2009; Geleynse et al., 2010; Hoyal and Sheets, 2009; Martin et al., 2009; Reitz et al., 2010), rather than lengthening of initiallyformed channels. To examine which conditions may support the formation of elongate deltas and whether these exhibit a signature of basin subsurface sediment composition, we performed two additional river-dominated delta simulations. Herein, all boundary and parameter settings were held the same; however, the initial width-to-depth ratio of the river was changed to obtain a virtually constant supply of fines through a relatively stable feeder channel (suppression of pattern formation; Struiksma et al. (1985)). Also, to notch this deeper channel into the shoreline, the initial basin depth was increased (basin slope was kept constant), to eliminate an otherwise initial imbalance between channel and basin bed levels. This changes the structure of the turbulent jet as well as reducing delta progradationaggradation rates. For both schematizations of the basin bed, the feeder channel still bifurcates at the river efflux, initially (Fig. 6a–b). For the case with fewer fines in the basin subsurface, the delta's morphology is characterized by a branching distributary network, entailing an approximate radially-extending deposit, analogous to the bifurcation-dominated experimental delta of Hoyal and Sheets (2009), formed by supercritical flow. Contrarily, for a basin subsurface which is relatively enriched with fines and being characterized by identical cohesive sediment transport rate and only slightly lower sand import rates and mean bed shear stresses at the river efflux (both groups are statistically quite similar; Fig. 6c), a predominantly single-paired distributary is formed, which clearly progrades via lengthening of its subaqueous levees, hence giving rise to a more elongated delta deposit. However, subsequent delta morphodynamics suggest a limit to this elongate delta-type. This limit is set by avulsion. For the latter case, while main distributaries still significantly prograde along their axii under decreasing τE and increasing Qrc (Fig. 6c), the previously-deposited mouth-bar and subaqueous levees are dissected (Fig. 6b). Resulting high-angle breaching has also been observed in real-world prograding channels (Rowland and Dietrich, 2005). Hence, as recently conjectured by Kim et al. (2009), a delta may feature multiple planimetric modes in its trajectory through time. This is also evident from the case with fewer fines in the initial subsurface,

223

where under decreasing τE and increasing Qrc (Fig. 6c), the previously merely distributive character of the delta gives way to significant elongation of relatively few distributaries (Fig. 6a). As such, characterization of deltas based on their planimetry at some point in time may be oversimplified, a point underlined by Bhattacharya and Giosan (2003). 3.5. Effects of tides There is another, yet unexplored mechanism which generates elongated delta channels. In the tide-influenced river delta simulations considered herein, riverine water level gradients and current velocities are modulated by a schematized tidal wave. Here, we restrict ourselves to presentation of results for a cross-shore propagating tidal wave, noticing that morphodynamics of simulated deltas for an alongshore-propagating tide are found to be surprisingly similar to those herein presented. In contrast to the effect of windgenerated waves, tidal action leads to most rough delta fronts, irrespective of subsurface sediment composition (Fig. 3, center panel). Under constant upstream water discharge, the large-scale longitudinal river water level gradient increases during falling tide (maximum at low water), whereas it decreases during rising tide (minimum at high water). Due to propagation into the confined and shallow river valley, the standing tidal wave is deformed, resulting in characteristic asymmetric riverine water level and current profiles; a prolonged ebb phase and a shortened flood phase. The outward-directed flow field at the river efflux is enhanced during the ebb phase and weakened during the rising part of the flood phase. Transport of fines towards the efflux is generally predominant over bed-load transport, irrespective of initial subsurface composition. Hence, note that, though the initial subsurface of the ‘sand-dominated sedimentary system’ only comprises sand, it eventually accommodates a mixed-load delta with largest riverine supply of fines (Fig. 2b). Again, this points at the importance of stratigraphic preconditioning. At initial stages of delta formation, distributaries form by incision at low water, while slightly aggrading during flood stage. This mechanism entails deposits of which the tidal signature can clearly be identified (Fig. 7a). With emergence of the delta distributary network, at low tide, ebb-

Fig. 6. Simulated planform dynamics of a river delta (a) and (b), here represented by the − 2 m mean water level contour, for two different initial schematizations of the river, and basin beds (50% (R4) and 25% (R5) sand by volume). (c) Relative (cohesive) silt flux (Qrc) and normalized excess shear stress (τE) versus simulated time (t/T, where T = 15 yr).

224

N. Geleynse et al. / Earth and Planetary Science Letters 302 (2011) 217–226

simulated delta mainly progrades via lengthening of a main center channel and initially-formed quite-regularly-spaced distributaries at its sides. While prograding, this center channel occasionally closes off one of its distributaries (Fig. 7b). For relatively increasing flood tide dominance (here, tide-induced flow reversals are limited to zones seaward of the delta front, due to a relatively high river discharge), this would potentially induce the transformation of a river-dominated distributary into a tide-dominated tortuous channel, supporting inferences from corings of the Mahakam delta (Storms et al., 2005), a well-known example of the mixed river-tide, fine-grained delta class (Orton and Reading, 1993). The simulated relative stability and elongated planform of tide-influenced distributaries are consistent with field observations (Olariu and Bhattacharya, 2006; Tanabe et al., 2003). 3.6. Linking the deposit to its feeder

Fig. 7. (a) Simulated tide-influenced river delta deposits for three initial schematizations of the sediment subsurface (100%–0%, 50%–50% and 25%–75% sand-silt by volume). The active stratal surfaces are shown. The sections are taken along a bar in the developing apex of the delta, after two tidal cycles, starting at high-tide level (HTL= +1.5 m MWL). Location of the sections is given by the dotted black lines in (b). Tidal signatures are not restricted to these sections, but also visible in sequences (including hiatuses due to erosion) elsewhere in the delta and river (Fig. 4c). (b) Corresponding simulated planform dynamics of a tide-influenced river delta after 4.5 and 7.5simulated years, here represented by the low-tide level contour (LTL= −1.5 m MWL). Note the dendritic structure of the channel network (Fagherazzi, 2008), elongated bars and seaward divergent planform of some (abandoned) distributaries. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

enhanced flow concentrates in the distributaries, leading to lengthening of their levees (Fig. 7b) and routing of fines that settle off the delta front (Fig. 2b). Incipient mouth-bar formation at termini of distributaries is observed (Fig. 7b), however, mouth bars generally do not persevere due to remobilization by the ebb-enhanced erosive jet. Contrarily, during flood stages, when the entire delta is submerged, fines are deposited on the proximal delta plain. Moreover, again, the effect of initial subsurface sedimentary schematization to delta morphodynamics can be observed. For an initially merely sandy basin subsurface, the delta progrades via lengthening of distributaries that grossly originate from the river efflux (delta apex). For a relatively high percentage of fines in the initial basin subsurface, the

Subsurface sediment composition not only controls characteristics of the delta deposit, it also controls morphodynamics of its feeder, the river. For initially merely sandy river beds and banks, multiple, relatively shallow channels form in a given cross-section (Fig. 3). Contrarily, for an increasing percentage of fines in the initial subsurface, a predominantly single, relatively deep channel forms. These channel patterns are similar across the different hydraulic forcing types. For all considered cases, the river valley widens () and attains a larger longitudinal slope (ζ) for increasing dominance of the sand fraction. Accordingly, a sand bed tends to accommodate a multiple-channel river that feeds its associated braidplain delta, whereas a cohesive fine bed tends to accommodate a main singlechannel river, which in turn, feeds relatively persistent sinuous channels that sweep across their delta-floodplain. This coupling is in striking general agreement with observations from nature (e.g., Makaske, 2001). It naturally follows that a river system forms an inherent structure with its downstream drainage basin; the feeder not only routes its upstream delivered sediments, but also transforms to take on characteristic planimetric-altimetric modes. It is well-known from geomorphologic fluid mechanics that fluvial style is a function of channel geometry, notably a channel's width-to-depth ratio (Seminara, 2006; Struiksma et al., 1985). To assess channel stability, sediment properties are often captured by a single representative parameter in morphodynamic models (e.g., a median grain diameter and via expressions such as a friction term and a degree of nonlinearity in the sediment transport formulation). This may turn out a generalized representation, most appropriate to unimodal sediment mixtures. Many natural river beds and emerged or submerged floodplains (deltas) host multiple principal physicalchemical behavioural particle classes that, once mixed, may, or may not, exhibit uniform behaviour to encountered fluid shear stresses. Coupled to bank and floodplain processes, including life or death, behaviour of sediment mixtures clearly provides an avenue for future research (Kleinhans, 2010). After all, it is the mobile interface between a fluid and its erodible boundary, rather than the flow itself that entails instability, which in turn, governs observed fluvial patterns (Seminara, 2006). Hence, understanding water flows and their boundaries, sediment subsurface, both contribute to understanding of dynamics and stratigraphy of rivers and their temporallyunconfined downstream extensions, i.e., their deltas. 4. Conclusions Here, we have shown the capability of a high-resolution physicsbased numerical model to accurately resolve the morphology and stratigraphy of developing river delta distributary networks under different hydraulic and sedimentary forcings, as known from numerous field observations, previously collapsed into a textbook

N. Geleynse et al. / Earth and Planetary Science Letters 302 (2011) 217–226

delta classification scheme. A new mechanism for the generation of elongated delta channels has been hypothesized, namely tidemodulated river outflow. This mechanism also entails cyclicity in delta-plain deposits. Moreover, our model study suggests the concept of stratigraphic conditioning in that different initial subsurface sediment compositions are assessed to result in distinctively different delta-plain morphometrics. Finally, our work points at the intrinsic coupling of deltas and their feeding rivers. Future work must focus on the significance of phasing of river discharge, wave, and tidal action to morphometrics and stratal patterns of deltas. Such episodicity in driving forces has been suggested to be important for several field deltas (Bhattacharya and Giosan, 2003; Fielding et al., 2005; van Maren, 2005). This also requires automatic extraction of the distributary network from gridded model output, a delicate problem, yet vital to accurately quantify metrics. Acknowledgments Funding was provided by Water Research Center Delft, Deltares and Statoil (contractno. 4501725117). The editor, M.L. Delaney, and referees are acknowledged for providing detailed and constructive feedback. References Allen, P.A., 2005. Striking a chord. Nature 434, 961. Andrews, D.G., McIntyre, M.E., 1978. An exact theory of nonlinear waves on Lagrangianmean flow. J. Fluid Mech. 89, 609–646. Ashton, A.D., Giosan, L., 2007. Investigating plan-view asymmetry in wave-influenced deltas. In: Dohmen-Janssen, C.M., Hülscher, S.J.M.H. (Eds.), River, Coastal and Estuarine Morphodynamics 5th IAHR Symposium, Enschede, The Netherlands, September 17–21, 2007. Taylor and Francis. Battjes, J.A., Janssen, J.P.F.M., 1978. Energy loss and set-up due to breaking of random waves. Proc. of 16th International Conference on Coastal Engineering, Am. Soc. Civ. Eng. (ASCE), Hamburg, Germany. Best, J.L., Ashworth, P.J., 1997. Scour in large braided rivers and the recognition of sequence stratigraphic boundaries. Nature 387 (6630), 275–277. Bhattacharya, J.P., Giosan, L., 2003. Wave-influenced deltas: geomorphological implications for facies reconstruction. Sedimentology 50, 187–210. Booij, N., Ris, R.C., Holthuijsen, L.H., 1999. A third-generation wave model for coastal regions. Part 1: model description and validation. J. Geophys. Res. 104, 7649–7666. Burrough, P.A., McDonnell, R.A., 1998. Principles of Geographical Information systems. Oxford University Press, Oxford. 333 pp. Chow, V.T., 1959. Open-channel hydraulics. McGraw-Hill College. Dingemans, M.W., Radder, A.C., de Vriend, H.J., 1987. Computation of the driving forces of wave-induced currents. Coast. Eng. 11, 539–563. Edmonds, D.A., Slingerland, R.L., 2007. Mechanics of river mouth bar formation: Implications for the morphodynamics of delta distributary networks. J. Geophys. Res. 112, F02034. Edmonds, D.A., Slingerland, R.L., 2010. Significant effect of sediment cohesion on delta morphology. Nat. Geosci. 3, 105–109. Edmonds, D.A., Hoyal, D.C.J.D., Sheets, B.A., Slingerland, R.L., 2009. Predicting delta avulsions: implications for coastal wetland restoration. Geology 37, 759–762. Edmonds, D.A., Slingerland, R.L., Best, J., Parsons, D., Smith, N., 2010. Response of riverdominated delta channel networks to permanent changes in river discharge. Geophys. Res. Lett. 37 (L12404). Eilertsen, R.S., Hansen, L., 2008. Morphology of river bed scours on a delta plain revealed by interferometric sonar. Geomorphology 94 (1–2), 58–68. Engelund, F., Hansen, E., 1967. A monograph on sediment transport in alluvial streams. Teknisk Forlag Kobenhavn, Denmark. Fagherazzi, S., 2008. Self-organization of tidal deltas. Proc. Natl Acad. Sci. USA 105, 18692–18695. Fielding, C.R., Trueman, J.D., Alexander, J., 2005. Sharp-based, flood-dominated mouth bar sands from the Burdekin river delta of northeastern Australia: extending the spectrum of mouth-bar facies, geometry, and stacking patterns. J. Sediment. Res. 75 (1), 55–66. Galloway, W.E., 1975. Process Framework for Describing the Morphologic and Stratigraphic Evolution of Deltaic Depositional Systems. In: Broussard, M.L. (Ed.), Deltas Models for Exploration. Houston Geological Society, Houston. Garcia, M.e.a., 2006. Sedimentation engineering: processes, measurements, modeling, and practice, vol. 110. ASCE. Geleynse, N., Storms, J.E.A., Stive, M.J.F., Jagers, H.R.A., Walstra, D.J.R., 2010. Modeling of a mixed-load fluvio-deltaic system. Geophys. Res. Lett. 37 (L05402). Gouw, M.J.P., Autin, W.J., 2002. Alluvial architecture of the Holocene Lower Mississippi Valley (U.S.A.) and a comparison with the Rhine-Meuse delta (The Netherlands). Sediment. Geol. 204, 106–121. Hasselmann, K., 1974. On the spectral dissipation of ocean waves due to whitecapping. Boundary Layer Meteorol. 6, 107–127.

225

Hasselmann, K., Barnett, T.P., Bouws, E., Carlson, H., Cartwright, D.E., Enke, K., Ewing, J., Gienapp, H., Hasselmann, D.E., Kruseman, P., Meerburg, A., Muller, P., Olbers, D.J., Richter, K., Sell, W., Walden, H., 1973. Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Dtsch. Hydrogr. Z. Suppl. 12, 1–95. Hoyal, D., Sheets, B.A., 2009. Morphodynamic evolution of experimental cohesive deltas. J. Geophys. Res. 114 (F2) F02009. Hutton, E.W.H., Syvitski, J.P.M., 2008. Sedflux 2.0: an advanced process-response model that generates three-dimensional stratigraphy. Comput. Geosci. 34, 1319–1337. Jerolmack, D.J., 2009. Conceptual framework for assessing the response of delta channel networks to Holocene sea level rise. Quat. Sci. Rev. 28 (17–18), 1786–1800. Jerolmack, D.J., Swenson, J.B., 2007. Scaling relationships and evolution of distributary networks on wave-influenced deltas. Geophys. Res. Lett. 34 (23), L23402. Kalkwijk, J., Booij, R., 1986. Adaptation of secondary flow in nearly-horizontal flow. J. Hydraul. Res. 24 (1). Kim, W., Jerolmack, D.J., 2008. The pulse of calm fan deltas. J. Geol. 116 (4), 315–330. Kim, W., Paola, C., Swenson, J.B., Voller, V.R., 2006. Shoreline response to autogenic processes of sediment storage and release in the fluvial system. J. Geophys. Res. 111 (F04013). Kim, W., Dai, A., Muto, T., Parker, G., 2009. Delta progradation driven by an advancing sediment source: coupled theory and experiment describing the evolution of elongated deltas. Water Resour. Res. 45 (W06428). Kleinhans, M.G., 2010. Sorting out river channel patterns. Prog. Phys. Geogr. 34 (3), 287–326. Kleinhans, M.G., Jagers, H.R.A., Mosselman, E., Sloff, C.J., 2008. Bifurcation dynamics and avulsion duration in meandering rivers by one-dimensional and three-dimensional models. Water Resour. Res. 44 (W08454). Lane, S.N., Richards, K.S., 1997. Linking river channel form and process: time, Space Causality revisited. Earth Surf. Processes. Landforms 22, 249–260. Lesser, G.R., Roelvink, J.A., van Kester, J.A.T.M., Stelling, G.S., 2004. Development and validation of a three-dimensional morphological model. Coast. Eng. 51, 883–915. Makaske, B., 2001. Anastomosing rivers: a review of their classification, origin and sedimentary products. Earth Sci. Rev. 53 (3–4), 149–196. Martin, J., Sheets, B., Paola, C., Hoyal, D., 2009. Influence of steady base-level rise on channel mobility, shoreline migration, and scaling properties of a cohesive experimental delta. J. Geophys. Res. 114, F03017. Nemec, W., 1990. Deltas remarks on terminology and classification. In: Colella, A., Prior, D.B. (Eds.), Coarse-Grained Deltas: International Association Sedimentological Special Publication, vol. 10, pp. 3–12. Olariu, C., Bhattacharya, J.P., 2006. Terminal distributary channels and delta front architecture of river-dominated delta systems. J. Sediment. Res. 76, 212–233. Orton, G.J., Reading, H.G., 1993. Variability of deltaic processes in terms of sediment supply, with particular emphasis on grain size. Sedimentology 40 (3), 475–512. Paola, C., Heller, P.L., Angevine, C.L., 1992. The large-scale dynamics of grain-size variation in alluvial basins, 1: theory. Basin Res. 4, 73 73. Paola, C., Straub, K., Mohrig, D., Reinhardt, L., 2009. The “unreasonable effectiveness” of stratigraphic and geomorphic experiments. Earth Sci. Rev. 97, 1–43. Reitz, M.D., Jerolmack, D.J., Swenson, J.B., 2010. Flooding and flow path selection on alluvial fans and deltas. Geophys. Res. Lett. 37, L06401. Ribberink, J.S., Blom, A., van der Scheer, P., van Straalen, M.P., 2002. Multi-fraction techniques for sediment transport and morphological modeling in sand-gravel rivers. In: Bousmar, Zech (Ed.), River Flow 2002, pp. 731–739. Roberts, H.H., Walker, N., Cunningham, R., Kemp, G.P., Majersky, S., 1997. Evolution of sedimentary architecture and surface morphology: Atchafalaya and Wax Lake Deltas, Louisiana (1973–1994). Gulf coast Assoc. Geol. Soc. Trans. 47, 477–484. Rodi, W., 1993. Turbulence Models and their Application in Hydraulics: a State-of-the art Review. Balkema, Rotterdam. Rodríguez-Iturbe, I., Rinaldo, A., 2001. Fractal River Basins: chance and Selforganization. Cambridge Univ. Press. Roelvink, J.A., 2006. Coastal morphodynamic evolution techniques. Coast. Eng. 53 (2– 3), 277–287. Roelvink, J.A., Walstra, D.J.R., 2004. Keeping it Simple by Using Complex Models. 6th Int. Conf. on Hydroscience and Engineering ICHE, Brisbane, Australia, May, 30-June, 3. Rowland, J.C., Dietrich, W.E., 2005. The Evolution of a Tie Channel. In: Parker, G., Garcia, M.H. (Eds.), River, Coastal and Estuarine Morphodynamics 4th IAHR Symposium, Illinois, U.S.A., October, 4–7, 2005. Taylor and Francis. Schumm, S.A., 1977. The Fluvial System. Wiley-Interscience. Seminara, G., 2006. Meanders. J. Fluid Mech. 554, 271–297. Seybold, H., Andrade, J.S., Herrmann, H.J., 2007. Modeling river delta formation. Proc. Natl Acad. Sci. USA 104 (43), 16804. Shaw, J.B., Mohrig, D.C., 2009. Incised bifurcations and uneven radial distribution of channel incision in the Wax Lake Delta, USA. In: AGU (Ed.), American Geophysical Union, Fall Meeting 2009 Abstract. 2009AGUFMEP41A0593S. Stelling, G.S., Leendertse, J.J., 1991. Approximation of Convective Processes by Cyclic AOI methods. In: Spaulding, M.L., Bedford, K., Blumberg, A., Cheng, R., Swanson, C. (Eds.), 2nd Int. Conf. on Estuarine and Coastal Modelling, Tampa, U.S.A., November, 13–15, 1991. Storms, J.E.A., 2003. Event-based stratigraphic simulation of wave-dominated shallowmarine environments. Mar. Geol. 199, 83–100. Storms, J.E.A., Hoogendoorn, R.M., Dam, R.A.C., Hoitink, A.J.F., Kroonenberg, S.B., 2005. Late-Holocene evolution of the Mahakam delta, East Kalimantan, Indonesia. Sediment. Geol. 180, 149–166. Struiksma, N., Olesen, K.W., Flokstra, C., de Vriend, H.J., 1985. Bed deformation in curved alluvial channels. J. Hydraul. Res. 23, 57–79. Stutz, M.L., Pilkey, O.H., 2002. Global distribution of deltaic barrier island systems. J. Coast. Res. 136, 694–707.

226

N. Geleynse et al. / Earth and Planetary Science Letters 302 (2011) 217–226

Swenson, J.B., 2005. Relative importance of fluvial input and wave energy in controlling the timescale for distributary-channel avulsion. Geophys. Res. Lett. 32 (L23404). Swenson, J.B., Paola, C., Pratson, L., Voller, V.R., Murray, A.B., 2005. Fluvial and marine controls on combined subaerial and subaqueous delta progradation: morphodynamic modeling of compound-clinoform development. J. Geophys. Res. 110 (F02013). Syvitski, J.P.M., Saito, Y., 2007. Morphodynamics of deltas under the influence of humans. Global Planet. Change 57, 261–282. Syvitski, J.P.M., Kettner, A.J., Overeem, I., Hutton, E.W.H., Hannon, M.T., Brakenridge, G.R., Day, J., Vörösmarty, C.J., Saito, Y., Giosan, L., Nicholls, R.J., 2009. Sinking deltas due to human activities. Nat. Geosci. 2, 681–686. Tanabe, S., Ta, T.K.O., Nguyen, V.L., Tateishi, M., Kobayashi, I., Saito, Y., 2003. Delta evolution model inferred from Holocene Mekong Delta, Southern Vietnam. In: Sidi, E.H., Nummedal, D., Imbert, P., Darman, H., Posamentier, H.W. (Eds.), Tropical Deltas of Southeast Asia-Sedimentology, Stratigraphy, and Petroleum Geology, SEPM: Special Publication, 76, pp. 175–188.

Thorn, C.E., Welford, M.R., 1994. The equilibrium concept in geomorphology. Ann. Assoc. Am. Geogr. 84, 666–696. Törnqvist, T.E., Kidder, T.R., Autin, W.J., van der Borg, K., de Jong, A.F.M., Klerks, C.J.W., Snijders, E.M.A., Storms, J.E.A., van Dam, R.L., Wiemann, M.C., 1996. A revised chronology for Mississippi River subdeltas. Science 273, 1693–1696. van Ledden, M., van Kesteren, W.G.M., Winterwerp, J.C., 2004. A conceptual framework for the erosion behaviour of sand-mud mixtures. Cont. Shelf Res. 24, 1–11. van Maren, D.S., 2005. Barrier formation on an actively prograding delta system: the Red River Delta, Vietnam. Mar. Geol. 224, 123–143. van Rijn, L.C., 1993. Principles of Sediment Transport in Rivers, Estuaries and Coastal Seas. Aqua publications, Amsterdam. 715 pp. Winterwerp, J.C., van Kesteren, W.G.M., 2004. Introduction to the Physics of Cohesive Sediment Dynamics in the Marine Environment. Elsevier. 576 pp. Wright, L.D., 1977. Sediment transport and deposition at river mouths: a synthesis. Geol. Soc. Am. Bull. 88, 857–868.