Home to hundreds of millions of people, major coastal cities and infrastructure, immensely productive wetlands, and some of the most compelling and diverse landscapes on Earth – yet low-lying and vulnerable to storms and rising sea levels – deltas are emerging as among the most critical environments in a changing world (Syvitski et al., 2009). They are also immensely complex. The science of deltas comprises, in roughly equal parts, geomorphology, ecology, hydrology, organic and microbial geochemistry, and human dynamics. The physical dynamics alone would present a formidable challenge, even if they were restricted to just turbulent flow interacting with sand; but most natural deltas involve major additional complications such as fine-grained cohesive sediment (mud) and strong, two-way interactions with biota.

A fundamental debate is developing across the sciences as to the best way to model and understand such complexity (e.g., Murray, 2003; Overeem et al., 2005; Paola and Leeder, 2011; Paola et al., 2011; Hajek and Wolinsky, 2012). Should we try to capture everything, creating models that simulate the processes in as much detail as current knowledge and computing power allow, or should we simplify, even at the risk of losing connections with reality? Modeling of deltas in recent years has produced excellent examples of both approaches, which we review below. Our aim here is to present a model that resides in the middle ground between detailed simulation and abstract simplification. We use a method based on weighted random walks, where the random walks are constrained by rules based on a hybrid of simplified governing equations for fluid motion and phenomenological representation of sediment transport processes. With suitable rules, DeltaRCM (reduced-complexity model for river delta formation) is able to produce delta morphologies that compare well with those produced by more complex models such as Delft3D and with the morphology of deltas in the field. We believe that the availability of abundant computing power strengthens rather than weakens the case for so-called reduced-complexity models such as the one we propose here. Understanding – as opposed to simulating – complex natural phenomena requires a spectrum of approaches and a clear understanding of the advantages and disadvantages of each.

The paper begins with a review (Sect. 2) of current approaches to modeling deltas, emphasizing previous reduced-complexity models. The detailed implementation of our model is presented in Sect.3, and results from it in Sects. 4 and 5. In Sect. 6 we discuss the meaning of the model results to date. Conclusions are provided in Sect. 7.

As with any morphodynamic model, the most direct delta formation model would solve the governing equations for water flow and sediment particles based on first principles, i.e., the conservation of mass and momentum or energy, in detail, given all the necessary initial and boundary conditions. However, this is still not practical, not only because of limits of computational power, but also because of the potential error accumulation in such complex “full physics” models (Hajek and Wolinsky, 2012). Existing models for delta formation cover a wide range of scales and complexity (Fagherazzi and Overeem, 2007; Paola et al. 2011).

On the simple side, models based on spatially averaged delta surface topography can predict average delta dynamics, such as laterally averaged surface profile, position of the shoreline, and position of the alluvial–bedrock transition (Parker et al., 2008; Kim et al., 2009; Lorenzo-Trueba et al., 2013). These models do not attempt to provide detailed structure, such as topography and channel networks. On the more complex side, to date, the most inclusive physics-based delta formation model is Delft3D, which solves a depth-integrated version of the Reynolds-averaged Navier–Stokes equations (shallow water equations) with a turbulence closure term for horizontal Reynolds stresses, and coupled with empirical sediment transport formulas based on bed shear stress (Lesser et al., 2004; Edmonds and Slingerland, 2007). Delft3D can resolve deltaic processes from smaller, engineering scales such as river mouth-bar formation and bifurcation (Edmonds and Slingerland, 2007) to larger, geological scales such as the whole delta morphodynamics controlled by sediment cohesion (Edmonds and Slingerland, 2009), waves, tides and antecedent stratigraphy (Geleynse et al., 2010). Delft3D is widely considered the best high-resolution delta model available to the research community, and its utility is greatly enhanced by the release of an open-source version in 2012. In the middle ground of the model complexity spectrum are the so-called reduced-complexity models (RCMs). These models feature descriptive constructions and intuitive simplifications over the hierarchy of natural processes, in contrast to highly detailed but computationally complex models such as Delft3D, while still evolving the topography and channel network without simplifying to the degree of spatially averaged models. The most common form of models in this category is a rule-based cellular routing scheme, such as the braided river model by Murray and Paola (1994, 1997) and some of the early erosional-landscape models (e.g., Willgoose et al., 1991). In terms of channel-resolving delta formation models, an excellent example is found in Seybold et al. (2007, 2009, 2010). In their model, the water flux field is calculated on a lattice grid via a set of simplified hydrodynamic equations which are equivalent to a diffusive-wave form of the shallow water equations with constant diffusivity. A few other examples of delta-related channel-resolving RCMs include an avulsive delta building model by Sun et al. (2002) and a channel-floodplain co-evolution delta building model, AquaTellUS, by Overeem et al. (2005).

RCMs are less computationally intensive than CFD (computational fluid dynamics)-based high-fidelity models yet still produce morphodynamic features at system scales, such as stream braiding and floodplain aggradation. While computational efficiency is often considered the reason for developing RCMs, their most important advantage is the flexible rule-based framework which allows for direct translation of phenomenological observations into the model (as opposed to hoping that they will emerge given a sufficiently detailed description of the underlying mechanics). The challenges of building a RCM for delta formation are the following: (i) the low topographic slope of the majority of river deltas (10-4–10-5) does not provide a strong guide for topographic flow routing, which is a key component in many RCMs for geomorphodynamic systems; (ii) the low slope together with relatively deep, slow channel flow creates a low-Froude-number environment such that the flow senses downstream information over relatively long distances, making it difficult to design localized rules which are essential for RCMs; (iii) the self-organized distributary channel network includes loops that further complicate flow routing; and (iv) many river deltas have suspended load and wash load as a primary sediment input component, which make sediment routing more complex than in a bedload-only system. In addition, the low-Froude-number flow condition implies, as the Froude number tends to zero, a “rigid-lid” condition in which the shape of the free surface is nearly flat. This condition potentially offers computational advantages as the flow depth can be estimated from a fixed surface elevation (usually sea level or a simple function using backwater equations) and bed elevation, but is almost decoupled from the bed topography.

In this work, we present a RCM delta model using the “weighted random walk” method. The basic goal is to develop a model that includes just enough of the dynamics to tackle the main problems listed above. To be more specific, we seek complexity-reduction in the following aspects: (i) the solution of water surface elevation, (ii) the flow momentum balance, and (iii) the criteria for sediment deposition and erosion. A detailed model description is given in the next section, followed by results and comparisons with experimental and field deltas, along with the results of more detailed delta models, and then a discussion of the strengths and weaknesses of our model approach.

DeltaRCM has two components: a cellular flow routing scheme as the hydrodynamic component, and a set of sediment transport rules as the morphodynamic component. The model uses a lattice of square cells for its domain, where water and sediment flux are routed in a cell-by-cell fashion. The model evolves in time by updating the depth-averaged flow field, water surface elevation, sediment flux, and bed elevation at each time step.

In this section we present various morphological features produced by DeltaRCM with different domain setup and input parameters. All simulations assume no effects from wave or tidal energy, i.e., the delta is a classic river-dominated delta (Galloway, 1975). We investigate (1) the effects of input sediment composition and (2) the model's ability to simulate deltas at field and laboratory scales. The former has been studied via field observation (Orton and Reading, 1993) and numerical simulation (Edmonds and Slingerland, 2009). The latter is based on the availability of data from experimental deltas; also, we believe that if a model can handle both field and experimental scales, it could potentially inform the interpretations and connections of both. Furthermore, we demonstrate DeltaRCM as a tool for hypothesis testing through study of the effects of the receiving basin depth.

As discussed above, two types of sediment are routed through the system: coarse (sand) and fine (mud). The ratio of the numbers of these two types of parcels at the inlet gives the ratio of sand and mud coming into the system. To set the physical scale of the simulation, domain and grid size are adjusted by changing cell size and physical input parameters, such as total input water and sediment discharge, and also global parameters such as the reference energy slope.

The input parameters (Table 2) include

the portion of sand in the upstream sediment input, fsand;

A delta writes (and rewrites) its own autobiography by building a sedimentary record from deposition and erosion. These sedimentary records allow us understand the past and to use delta deposits to reconstruct their range of natural behavior. Therefore, the ability to record stratigraphy in a delta formation model enables us to directly investigate the connection between surface and subsurface processes. In this model, we have two methods that track the stratigraphy of model-produced deltas: the first method tracks the distribution of coarse and fine sediment by recording the percentage of sand in each deposition event; and the second method tracks the age of the deposit by labeling each deposition event with the time that its sediment enters the domain from the inlet channel.

To track stratigraphy each cell in the domain is viewed as a storage column (shown in Fig. 2), and the volume below the bed surface is further divided into thin layers of an equal thickness (these layers are visible especially in Figs. 11 and 12). The thickness is chosen to be about a thousandth of the reference depth, although it can be set to different values to allow for different vertical resolutions. Each layer is recorded with a value associated with it – at present it is either the percentage of sand (a value between 0 and 1) or the age of the deposit (represented by the number of time step). For example, if a cell has net deposition, the volume it received from passing parcels will fill up as many layers as needed above the previous bed surface, and all values associated with these layers are set to the ratio between the volume of sand deposited and the total volume of sediment deposited during this time step. If a cell has net erosion, the bed surface will be lowered and all values associated with the layers above the new bed surface will be erased (by resetting these values to -1 in the code).

Here we present two examples. (1) We take a sample run of a field-scale delta and 30 % sediment input (run 4). In Fig. 11, we show a stratigraphic slice in the dip direction along the center line of the inlet channel. In Fig. 12, we show the time series of the stratigraphic slice in the strike direction about 20 cells (1 km in this case) away from the inlet channel. In both figures white represents pure sand and dark blue represents pure mud, with mixed deposits represented by linear combinations of the two endmembers. Generally speaking, coarse sediment (sand) can be found in channel belts and mouth bars, while fine sediment (mud) can be found in distal regions such as the bottom set of the delta, on the floodplain or in abandoned channels. (2) In Fig. 13 we show a sample model run for laboratory conditions (run 8). Note the evolution of the area pointed to by the yellow arrow. The series of images shows the deposition sequence from an individual avulsion event.

One of the themes running through this paper is that even in the framework of a reduced-complexity delta model there are a number of important details that must be modeled fairly accurately to achieve even qualitatively correct model results. These include a reasonably accurate representation of the water surface and the inclusion of suspended sediment deposition and entrainment. To demonstrate the importance of the water surface we switch off this component in routing water parcels, i.e., we set the partitioning coefficient (γ) to zero. In this case, the delta is completely dominated by inertia and as a result a single elongated channel extends without avulsion or bifurcation (Fig. 14a). (Note that this is not the same behavior as setting the input sediment to contain 0 % sand which exhibits multiple elongated channels – see Fig. 14b.) By contrast, the effect of deposition and entrainment of fine-grained sediment in DeltaRCM is illustrated by removing the suspended sediment load from the calculation. In such a case, the predicted channels are highly mobile and levees separating channels and floodplains are absent; i.e., we arrive at a delta formation with no stable channel networks, the characteristics of an alluvial fan (Fig. 14c). The importance of the water surface and suspended sediment is also well illustrated in the previous RCM delta model developed by Seybold et al. (2007, 2009), where a reduced-complexity water surface and depth calculation, along with a treatment of cohesive and noncohesive sediment behaviors through a flow strength and flow velocity terms, respectively, was able to build both elongated bird-foot and multichannel fan deltas. Part 2 of this work further explores the hydrodynamic mechanism of the water surface and investigates the feedback between the flow solver and the sediment transport processes in determining channel bifurcations.

The need for accurate representation of some of the physical details in DeltaRCM is quite striking compared to the success of even fairly radical reduced-complexity approaches in modeling other morphodynamic environments such as erosional landscapes (e.g., Willgoose et al., 1991), braided rivers (e.g., Murray and Paola, 1994) and eolian bedforms (e.g., Werner, 1995). So why is it that deltas seem to require more attention to detail? Can we learn anything from this experience that might help us better understand what systems are most and least amenable to reduced-complexity approaches?

Since deltas and drainage basins share dendritic channel patterns – one is a distributary network while the other is a tributary network – we first look at the differences between these two systems. In modeling the evolution of drainage tributary networks, even highly simplified relations for water flux and sediment transport yield quite reasonable drainage networks and elevation changes in the long-term evolution of catchments (e.g., Willgoose et al., 1991). The equation describing the evolution of land elevation in Willgoose et al. (1991) includes two transport processes: fluvial transport and diffusive transport. The former is dependent on the discharge and the slope in the steepest downhill direction, and the latter is dependent on slope and diffusivity. Relations of similar simplicity cannot be easily applied to modeling deltas because deltas are low-gradient environments where the transport direction and capacity are to some extent decoupled from bed elevation and slope. To be more specific, (1) bed slope in low-gradient environments is often uncorrelated with flow direction and strength; for example, bed slope points opposite to the direction of flow where channels shoal up towards the shoreline; (2) the water surface, which dominates local flow routing, is largely independent of bed topography; (3) the typical low-Froude-number flow in low-gradient deltaic environments creates strong backwater effects that imply strong nonlocality in flow and sediment flux control (Lamb et al., 2012; Nittrouer et al., 2011) – meaning that downstream conditions control upstream flow dynamics (Hoyal and Sheets, 2009); and (4) river mouth and shore processes such as waves and tides also control the overall morphology of deltas, providing additional process complexity.

According to Werner (1995), for a nonlinear and dissipative system, considerable simplification can be applied if the system exhibits the following two properties: (1) it has a finite number of steady states as “attractors”, and (2) it has macroscopic emergent behaviors that are self-organized and consistent with, but decoupled, from microscopic physics. If we compare drainage networks with deltas, the former exhibits a strong generic pattern and scale-invariant properties expressed in generalizations such as Horton's laws (Horton, 1945). In contrast, the networks on deltas have many varieties, responding to a wide range of processes; no universal geometry applies to them all. Regarding model complexity, the lack of universality in the system pattern indicates the requirement for a more detailed, system-specific approach in modeling them.

So, is the low gradient the main cause of the modeling difficulty, making deltas more “unforgiving” than erosional landscapes in terms of the accuracy of hydrodynamic calculation? For cellular models that use explicit flow routing schemes, the complexity level rises as factors other than topographic slope alone determine water and sediment routing. It also increases with nonlocality in the broad sense of the sensitivity of dynamics at one point to conditions far away in the system. Other contributing factors such as water surface gradient and flow inertia weigh in as the overall topographic gradient decreases. For example, dune fields may have very low to zero average topographic slope, but they have locally high steepness meaning that, as in erosional landscapes, the sediment dynamics are dominated by bed topography. In deltas, however, the controlling factor is the relatively subtle water surface topography, therefore simple descriptions relating sediment deposition and erosion to e.g., local elevation and slope give realistic dune field dynamics but do not work in deltas.

Can we be more systematic about evaluating the amount of detail needed to model a geomorphodynamic system? This is an important fundamental question in morphodynamic modeling, and we do not pretend to resolve it here. But our experience with DeltaRCM suggests the following guidelines as a starting point.

For gravity-driven systems, the overall gradient of the landform is one important index in the sense that in high-gradient systems the gradient alone is enough to route the flow.

In this paper we have introduced a new reduced-complexity model (RCM) for river delta formation. Key techniques include that (1) water and sediment fluxes are represented as parcels and routed through the domain in a Lagrangian point of view; (2) the movements of parcels are based on a probability field calculated from rules abstracting the governing physics; (3) deposition and erosion are achieved by exchanging the volume of passing sediment parcels and bed sediment columns, and the condition for this exchange depends on a set of rules that distinguish bedload and suspended load; (4) bed sediment columns record the composition of coarse and fine material in layers; (5) a topographic diffusion process takes into account cross-slope sediment transport and bank erosion. By varying input conditions such as the ratio of coarse and fine sediment, reference slope, and dimensions of the domain, the simulated deltas yield a range of different behaviors that compare well to higher-fidelity model results and observations of field and experiment deltas.

We find that the relatively simple cellular representation of water and sediment transport is able to replicate delta morphology at the scale of channel dynamics, including the emergent channel network with channel extension, bifurcation and avulsion. Here, we summarize the basic components needed for a RCM to produce major static and dynamic features of river deltas:

a depth-averaged flow field that guides sediment transport