In this work we develop a reduced-complexity model (RCM) for river delta formation (referred to as DeltaRCM in the following). It is a rule-based cellular morphodynamic model, in contrast to reductionist models based on detailed computational fluid dynamics. The basic framework of this model (DeltaRCM) consists of stochastic parcel-based cellular routing schemes for water and sediment and a set of phenomenological rules for sediment deposition and erosion. The outputs of the model include a depth-averaged flow field, water surface elevation and bed topography that evolve in time. Results show that DeltaRCM is able (1) to resolve a wide range of channel dynamics – including elongation, bifurcation, avulsion and migration – and (2) to produce a variety of deltas such as alluvial fan deltas and deltas with multiple orders of bifurcations. We also demonstrate a simple stratigraphy recording component which tracks the distribution of coarse and fine materials and the age of the deposits. Essential processes that must be included in reduced-complexity delta models include a depth-averaged flow field that guides sediment transport a nontrivial water surface profile that accounts for backwater effects at least in the main channels, both bedload and suspended sediment transport, and topographic steering of sediment transport.

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

Illustration of the basin, boundaries and inlet channel.

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.

The physical setting of our delta formation model is simplified to a
rectangular basin of constant water depth (

Diagram of the lattice grid and the primary values at each cell (water unit discharge, water surface elevation and bed elevation). Note that the total number of cells is reduced for the illustration.

For water and sediment routing, we first define a set of global parameters
that remain constant for each model run: (1) a reference water depth

The domain is shown in Fig. 2, with cell size

Two types of parcels that carry a water or sediment attribute are routed
through the domain. A time step is defined by the addition of

Within each model run, the size of the time step

The operations can best be understood by describing the processes in a single time step. There are four distinct phases: (1) the addition and routing of the water; (2) updating of the water surface elevation; (3) routing the sediment parcels and updating the bed elevation through deposition and erosion; and (4) updating of the routing direction, a vector field that determines the direction of flow through each cell in the domain. Each of these phases is described in turn.

To prepare, we divide the upstream water discharge (

At the start of a time step we assume that we have a delta with known shape
and topography, i.e., at each cell we have a value of the water surface
elevation

For the purpose of routing water, we define a binary cell state: 0 – dry,
1 – wet. This is done by doing a sweep through the domain and marking cells
with a water depth larger than a small threshold value

The process in the first part of the time step requires us to route, in turn, each of the water parcels through the domain. When the parcel is at a given cell, a decision is needed indicating to which of the eight neighbor cells it will move to. We achieve this by using a so-called weighted random walk where the movement is dictated by a predefined probability distribution between the eight neighbor cells. The specification of the probability distribution is as follows.

At a given cell, first we calculate the routing weights for the eight neighbor
cells. With the local routing direction

The weights above are calculated only for the wet neighbor cells of the given
channel cell. All dry neighbor cells take a weight value of 0. At each
channel cell we can then calculate routing probabilities

Then, for the purpose of later sediment transport, we need to estimate the local
flow unit discharge and velocity. To do this we take the cell discharge
vector (m

Water surface elevation is essential in this model not only because it
participates in the calculation of flow depth but, even more importantly, because the
gradient of water surface plays a major role in determining the routing
probabilities,

In this reduced-complexity model, our goal is to obtain a sufficiently accurate surface profile without solving the full 2-D hydrodynamic equations. We propose a method that uses a finite-difference scheme along the movement path of individual water parcels, analogous to the simplified surface solver developed by Rinaldo et al. (1999).

Definition of cellular direction

Calculation of the direction of the cell-representative discharge
vector. The representative discharge vector takes the direction of the
summation vector of all contributions from each visiting water parcel, and
for

A diagram showing the path of one individual water parcel compared to smooth flow streamlines.

To start with the simplest formulation, we assume that water surface slope
along a channel streamline can be approximated by the reference slope

First, we need to locate the part of the path that is on the delta surface, as the part in ocean is considered flat. In general, a water-parcel path starts at one of the inlet cells, moves from one cell to an adjacent cell, and ends at one of the downstream ocean boundary cells. We distinguish the cells along the path on the delta surface and the cells in the open ocean by checking two values at each cell such that either a cell is on the delta, or a cell is in the ocean if both of the following criteria are met:

local bed elevation

local flow speed

With a given water-parcel path, the calculation starts from the end of the
path and goes backward towards the inlet. For the

if cell

if cell

This calculation gives the surface profile along the path of an individual
water parcel and is repeated for all water-parcel paths. There are two
additional situations to be taken care of.

If a cell is visited by multiple water parcels, all the values from each visiting path are recorded and an average value is taken from these stored values in the end to obtain a single value for water surface elevation at each cell.

If a cell is not visited by any water parcels, its water surface elevation retains the old value (from the previous time step).

In the end, the water surface elevation is updated with an underrelaxation
scheme for numerical stability:

To ensure conservation of water mass, the unit discharge field remains the same within one time step. Therefore, as the water surface elevation is updated, only water flow depth and velocity are adjusted accordingly.

Now, both the flow field,

coarse sediment, referred to as “sand”, is noncohesive, and transported as bedload;

fine sediment, referred to as “mud”, is cohesive, and transported as suspended load.

For routing sediment parcels we use the same weighted random walk method as
for the routing of water parcels (Eq. 6) with two modifications:

The routing direction

Transport resistance for sediment maintains the inverse function of flow
depth but has different exponents. The idea is that sediment flux tends to
concentrate in the lower portion of the water column and therefore it is more
likely to follow topographically low areas. For now we use an exponent

Sediment parcels are routed sequentially in a weighted random walk fashion according to the probabilities calculated with Eqs. (13), (14) and (15). The change to the bed topography is obtained by the exchange of sediment volume between the moving parcel and the local bed at each cell along the path – during deposition a sediment parcel loses part of its volume and this volume is added to the bed, and vice versa for erosion. We use simple phenomenological rules to decide (i) where deposition or erosion happens and (ii) how much volume should be exchanged between the sediment parcel and the bed. The rules for sand and mud parcels are different.

For convenience of description, we refer to the initial volume of each
sediment parcel

For the deposition from a sand parcel we do the following:

At each cell in the domain, we calculate a “transport capacity” for
sand flux,

Similar to the calculation of water discharge, as the sand parcels are
routed sequentially, we track the accumulated total sand flux,

Deposition occurs if a sand parcel visits a cell that has an accumulated
local sand flux exceeding the transport capacity:

For deposition from a mud parcel we do the following:

Deposition occurs if a mud parcel visits a cell that has a local flow
velocity

Erosion occurs if local flow velocity magnitude is larger than a threshold
value,

For a sand parcel,

For a mud parcel,

At each step, the volume of the sediment parcel is updated as

The elevation of the local bed is updated as:

The local flow velocity and flow depth are updated in accordance with
each event of deposition or erosion:

The reason for updating local flow depth and velocity immediately after each event of deposition and erosion is to avoid excess change to the bed. Similarly, we add an extra control on the rate of change to the bed by limiting the amount of deposition and erosion by a sediment parcel so that the change to local depth is less than 25 %, so that the change to local flow velocity is less than 33 %. For example, if local flow depth is 4 m, then the maximum deposition or erosion by a single sediment parcel is limited to 1 m change to the bed.

After all sediment parcels finish their random walk, to take into account
the influence of topographical slope on sediment flux in an approximation of
the Bagnold–Ikeda expressions (García, 2008), we apply a topographic
diffuser that assumes the diffusive flux is proportional to local sand
(bedload) flux and topographical slope:

Before moving to the next time step, we need to update the routing
direction: a unit vector at each cell indicating the downstream direction
for routing water parcels. In this last phase of the time step, at each cell
we calculate the updated value of the unit water discharge vector

To achieve this, we combine two physical processes dictating the flow direction: (i) at an instant in time flow has a tendency to continue in the same direction as the direction at the previous instant due to inertia, and (ii) in the absence of any other drivers the flow goes downslope which in our case is indicated by the water-surface slope rather than bed slope.

List of model constants and parameters.

First, we calculate a unit vector from the downstream direction based on the
previous time step:

By implementing the method described in this section, we have achieved our goal of complexity reduction: (i) the construction of the water surface via 1-D profiles captures the overall trend of water surface gradients without solving the full hydrodynamic equations; (ii) the flow momentum balance is relaxed, e.g., the effect of flow inertia is considered only in the form of direction rather than magnitude; and (iii) the criteria for sediment deposition and erosion are in the very basic form of a nonlinear relation between sediment carrying capacity and flow velocity. Key constants and parameters that do not vary in our tests are listed in Table 1. In the next section, we will show that when implemented in our DeltaRCM model these reduced-complexity constructions predict delta growth characteristics and channel dynamics that are comparable to those of high-fidelity modeling and field observations.

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,

global parameters; i.e., the reference flow depth

total discharge

Time series of delta formation with different ratios of sand and mud flux (runs 1, 2 and 3). The time interval between rows is roughly 200 days of delta building time with continuous bank-full discharge.

In this group, the domain is a lattice grid of 120 by 60 square cells. Cell
size is taken to be 50 m. The channel inlet is five-cells wide (250 m), with a
reference flow depth of

Comparing shoreline roughness between simulated deltas with input sand fractions of 25, 50, and 75 %. Shoreline roughness here is measured by the ratio between (i) the number of cells in the domain that contain a piece of shoreline of the simulated delta, and (ii) the average radius of the delta toposet in number of cells.

List of delta model runs and parameter values.

We show three model runs in Fig. 6 with the portion of sand in the upstream
sediment discharge set to 25, 50, and 75 %. The resultant deltas
differ systematically based on the input mud fraction in the following
characteristics, which are consistent with those found in the investigation
on the effects of sediment cohesion by Edmonds and Slingerland (2009).

On a sandy delta the channels are relatively shallow and mobile, without well-defined levees. Flow is less confined. There are large areas of sheet flow. The shoreline is smooth and the delta grows roughly in a semicircular shape.

On a muddy delta, channels are deeper and stable, with well-defined levees. Channels tend to elongate. The shoreline is rugose, and deltas build in different directions by switching lobes.

The contrast in the model-predicted roughness between a sandy and muddy
delta is illustrated in Fig. 7, where plots of the time variation of the ratio
of number of cells on the shoreline to average delta radius (measured in number
of cells) is presented. In these calculations, the shoreline is defined using
the opening-angle method (OAM) developed by Shaw et al. (2008), employing an
elevation threshold of

Laboratory experiments, numerical modeling and field observation are three
important approaches of understanding the formation of deltas. Because we would like
to test our model across as wide a scale range as possible, we include
experimental deltas at laboratory scales. To do this, we change the domain
to a lattice grid of 90 by 180 cells with a cell size of 0.02 m. The inlet
channel is still five-cells wide but has a flow depth of 0.02 m and a water
discharge of 0.6 L s

In Fig. 8a–f we show a time series of the resultant deltas during one avulsion cycle. These plots reveal the key characteristics of an alluvial fan delta, in which a few active channels quickly switch (avulse) to build a semicircular shape with a relatively smooth shoreline (Reitz and Jerolmack, 2012). To evaluate the details of this channel-switching process, we calculate the wet fraction of delta surface that is covered by active channels (defined by cells that have a flow velocity greater than 50 % of the characteristic flow velocity) and plot it against time (Fig. 8g). Each avulsion event can be identified by a sudden drop of the wet fraction followed by a relatively slow rise caused by backfilling and flooding. An avulsion timescale estimated from this plot is in the range of 5–10 min, a value that is of the same order as the laboratory observations made by Reitz and Jerolmack (2012).

The series of images matches the avulsion cycles observed in
physical experiments (Reitz and Jerolmack, 2012):

It has been suggested that the accommodation – the space that a delta can grow into – plays an important role in the architecture and behavior of a growing delta (e.g., Paola, 2000; Heller et al., 2001). However, for the case of river deltas with very low-Froude-number flow, it is still unclear how the depth of the basin affects the overall morphology of the delta. Storms et al. (2007) use Delft3D to model initial delta formation from a river effluent discharging constant flow and sediment loads into shallow and deep receiving basins under homopycnal conditions; they show that the shallow basin delta is dominated by mouth-bar bifurcations and a shoaling channel network, and exhibits significant stratigraphic complexity and subaerial development, while the deep basin delta is dominated by unstable bifurcations, levee breaches and avulsions (Storms et al., 2007). The authors suggest that the shallow basin case resembles the Wax Lake Delta. In our model runs 6 and 7, we test scenarios with the same inlet channel conditions and discharge, but different basin depths. In run 6, the receiving basin depth is half of the reference depth (defined by the inlet channel which is supposed to be at equilibrium state in terms of sediment transport), while in run 7, the receiving basin depth is double the reference depth. In Fig. 9 we show that our results yield similar behaviors to the ones modeled by Storms et al. (2007) using Delft3D. For the shallow basin the morphological development is very close to the description of Storms et al., while the deep basin delta has similar outcomes but the middle ground bar and avulsion over the levee are not as clear in the RCM results.

Two model runs (runs 6 and 7) with different basin depths and everything else the same. The shallow basin delta is dominated by more frequent bifurcations while the deep basin delta is dominated by few channels with more avulsions.

The differences between a shallow and deep receiving basin, according to our
model results, are the following:

Channels will still try to maintain the same unit power of transporting sediment by maintaining a certain cross-sectional geometry with levees on the side and erosion or deposition on the bottom.

In general, a distributary channel network shoals up and channels are stable at shallower depths going seaward. With a shallow basin the amount of work is reduced. Also, the narrow space promotes the splitting of flow which enhances the growth of a distributary network.

A deep basin increases the timescale of establishing a stable channel and, therefore, introduces stronger competition among channels by allowing larger differences to develop.

The total number of active channels is higher in the shallow basin case, with about 5–6 channels, as compared to 1–3 channels in the deep-basin case.

The subnetwork mainly collects fine sediment from the main channel network, which requires a much slower flow to settle.

As the tributary subnetwork joins into bigger trunk channels, the ability of the flow to carry sediment increases.

Finally, at the downstream end of the network, where the trunk channel collecting water coming out of the island meets the open water, the sorting of the sediment deposited is very similar to a normal channel that has a coarser bar-like structure at the mouth.

Flow features on the island of a delta formed in a shallow basin.

Stratigraphy slice in the dip direction of run 4 (30 % sand input). Note the layering of coarse and fine grains over time. Yellow arrow points to the bottom layer that accumulates fine grains at the bottom set of the delta; orange arrow points to the coarse grain layer deposited by channels that used to be active at that location; the two together show the classic “coarsening-up” pattern in stratigraphy. The red arrow points to the fine grains deposited after the channels are abandoned.

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

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.

Time series of the stratigraphic slice in the strike direction
about 20 cells (1 km) away from the inlet channel. Note that between

Time series of a delta produced by DeltaRCM with laboratory
settings, and stratigraphy slices in the strike direction about 20 cells
(0.4 m) away from channel inlet. Note the evolution of the area pointed to by
the yellow arrow.

Effects of model parameters.

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 (

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.

A closely related indicator is the wetted area fraction in the sense that a combination of low wetted fraction and high topographic gradient is the limit in which steepest-path methods (Passalacqua et al., 2010) are sufficient to determine the flow path, without the need for simulation of the flow details.

Froude number (

For systematic behaviors on scales greater than the backwater length scale, in-channel-scale hydrodynamic details can be resolved at much lower complexity; this applies for example to avulsion models that use single-cell-wide threads to represent channel belts (Jerolmack and Paola, 2007).

Whether the system to be modeled exhibits a strong generic pattern or scale-invariant (e.g., fractal) properties, the lack of universal patterns in a dynamic system is an indicator of sensitivity to local detail.

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

a nontrivial water surface profile that accounts for backwater effects at least in the main channels

representation of both bedload and suspended load

topographic steering of sediment transport.

the instability at channel mouths that creates bars and subsequent bifurcation

the variation in water surface profile associated with lobe extension that causes channel avulsion

water surface slope along channel sides which creates flooding onto the floodplain.

This work was supported by the National Science Foundation via the National Center for Earth-surface Dynamics (NCED) under agreement EAR-0120914 and EAR-1246761. This work also received support from the National Science Foundation via grant FESD/EAR-1135427 and from ExxonMobil Upstream Research Company. The authors thank P. Passalacqua, D. A. Edmonds, N. Geleynse, and J. Martin for discussions and comments. The authors also thank S. Castelltort, R. Slingerland and A. Ashton for their insightful comments and reviews. Edited by: S. Castelltort