In this work, we simulate barchan swarms using the Two-Flank Agent-Based Model and investigate how changes to model parameters and environmental drivers lead to different swarm dynamics. In particular, we explore how the parameter

Crescent-shaped barchan dunes are one of the most striking aeolian bedforms and are found on Mars

Individual barchans are typically tens of metres wide

Swarms display many emergent phenomena such as size selection

Not only are the length scales of swarms orders of magnitude larger than those of the dunes themselves, but the timescales over which these systems evolve are also much longer than for individual barchans

Several ABMs have been implemented to study barchan swarms

The earliest barchan ABM

We have developed a new barchan ABM, the Two-Flank Agent-Based Model (TFABM), which has been shown to not only reproduce all of the observed collision behaviour of barchans but which can also be used to simulate the formation of asymmetric bedforms under the influence of variable wind regimes

The TFABM and its capabilities to replicate all known barchan collision dynamics were introduced in

The major difference between the TFABM and previous barchan ABMs is the structure of the dunes (agents) themselves. In the TFABM, each dune is represented by its two flanks, which are able to change size semi-independently of one another. The position of the dunes is recorded as the coordinates of the upwind toe. The central axis of the dune extends in the direction of the dominant wind, which we define to be the positive

The width of each horn is a linear function of the flank width

Unlike previous barchan ABMs, the more realistic morphology of dunes in the TFABM allows for variation in the direction of wind. Incoming sand flux is absorbed by each flank across the windward projection of its width. For oblique winds, this means that the windward flank will absorb proportionally more of the flux, starving the leeward flank of material.

As well as absorbing material across their wind-facing width, dune flanks also lose material across the width of their horns. In the initial study in which the TFABM was introduced, flux was emitted from the horns at a saturated rate, regardless of the influx the horn received, in line with findings from continuum simulations

Finally, in asymmetric dunes, there is an effective transfer of material from the larger to the smaller flank. If this were not the case, then any asymmetry in a barchan would persist, which contradicts with the view of symmetric barchan morphology as an attractor ^{2} yr^{−1}, which is the average value of those reported for Tarfaya in

The sizes of the flanks of a dune are coupled only through the third term in Eq. (

For highly asymmetric dunes, the typical barchan geometry shown in

Thus, combining these two terms, the maximum asymmetry of a dune with flank widths

One of the reasons that barchans have been studied so extensively is that they migrate rapidly at a rate governed by the sand flux conditions and the size of the bedform. We use the commonly accepted expression for the migration rate as

From this expression, it follows that larger dunes will migrate more slowly than smaller ones, which ensures that collisions are frequent occurrences in barchan swarms. The TFABM uses a simple rule for collisions, which is, nonetheless, able to reproduce a wide range of known collision behaviours

The intersecting flanks merge together.

The volume of merged flanks is used to calculate an effective flank width

A non-intersecting flank with the width

If

Finally, the flanks which merged together form a barchan with a preserved centre of mass and an asymmetry ratio determined by the relative mass that was initially to the left and right of the centre of mass.

The algorithm itself can produce between one and three output dunes, although calving may occur on the following time step, such that a collision may effectively produce up to four outputs. We refer to collisions which decrease the number of dunes (i.e. have a single output) as merging or aggregation, collisions which conserve the dune number as exchange, and collisions which increase the dune number as fragmentation. For more information on the phase space of the TFABM collision rule, see

The simulation space has a downwind length of 10 km and a central strip with a width of 3 km or, in some cases, 5 km set to ensure a sufficient number of dunes were present. On either side of the central strip there were empty strips of the same width to allow for an oblique movement of the dunes. At the upwind boundary over the entire width (9 or 15 km) of the simulation space, we supply a free flux

In this work, we do not use periodic boundaries; thus, once dunes exit the simulation space, they are lost. To account for this, new dunes enter the simulation space at the upwind boundary. The rate of injection of new dunes is governed by assuming that just upwind of the domain, there is a number density

For the first series of simulations, we use a Gaussian distribution for the direction of the wind centred around 0°. We experimented with different values of the standard deviation but found that, provided the standard deviation remains relatively small (we tested up to 5°), the exact value is not important. Therefore, all of the simulations shown in this section were performed using a standard deviation of 3°. Since the wind angle is normally distributed, this standard deviation ensures that 99.7 % of the time, the angle of the wind is within

As shown in ^{−2} and performed simulations with

The final states of the simulated swarms and plots showing that the mean dune width and population size had stabilised for the unscaled outflux, ^{−2}.

The final states of the simulated swarms and plots showing that the mean dune width and population size had stabilised for the scaled outflux, ^{−2}.

Regardless of the choice of outflux, lower values of

Relative frequencies of different types of collisions: merging (circle), exchange (triangles), and fragmentation (squares) observed in simulations with ^{−2}.

It has been found that the sizes of dunes in real-world swarms remain stable in the longitudinal dimension

The mean width and dune density in 500 m cross sections averaged from measurements at the end of each year once the swarm properties stabilised.

Figure

The mean width and dune density in 500 m cross sections averaged from measurements at the end of each year once the swarm properties stabilised.

Whereas, in some instances, we observe longitudinally homogeneous dune sizes, the simulated swarms are not homogeneous in the dune number density. In all of the simulated swarms with unscaled outflux, the overall dune density shows a decreasing trend with downwind distance. On the other hand, for the scaled outflux, the dune density initially increases to a maximum and then begins to decrease in all cases. However, for

We have already shown how varying

Histograms of dune asymmetries defined at the ratio of port and starboard flank widths are shown in panels ^{−2} and scaled flux. The dotted black lines show the distribution from a real-world swarm in Tarfaya.

The standard deviation of the ratio of flank widths for (

The previous section shows that varying ^{−2} in those simulations. Real-world swarms, however, are observed to have a wide range of densities ^{−2} (these correspond to 5%, 10%, and 15% of the density of Tarfaya 1 in

The final states of simulations and stabilisation of the dune number and mean size with time for swarms with scaled outflux. ^{−2} in panels

The longitudinal variation in the dune size and the number density are shown in Fig.

The mean width and dune density in 500 m cross sections averaged from measurements at the end of each year once the swarm properties stabilised. The grey areas represent 1 standard deviation. Scaled outflux was used for simulations with ^{−2} in panels

As we have already discussed, for ^{−2} the dune density exhibits a non-monotonic behaviour with downwind distance, but after a peak at around 2.5 km, the number density decreases, moving downwind. The same trend is observed for an injection density ^{−2}, with the peak moving downwind to around 3.5 km. However, for ^{−2}, we are able to produce a swarm which for which the dune density remains constant from around 4.5 km. In the same swarm, the size distribution stabilises at around 6 km, meaning that over the final 4 km of the simulation space both the dune density and dune size are homogeneous, as expected for real-world swarms. The size distribution of that swarm from 6 km (when it stabilises) is shown in Fig.

The stable size distribution from 6–10 km in the scaled flux simulation with ^{−2}. The solid black line shows a fitted log-normal distribution.

Recent studies have begun to consider the impacts of future anthropogenic climate change on dune systems

To explore how the TFABM might be used to study such human-induced changes, we performed simulations in which a swarm was allowed to form and stabilise under a particular combination of boundary conditions before these conditions were then suddenly changed.

In Fig.

A simulated swarm and its longitudinal width profile at 300, 400, 500, and 600 years in panels ^{−2} until 356 years, after which ^{−2}. The dashed and dotted lines represent the results from simulations where ^{−2} respectively, with the other model parameters unchanged.

Although there is no long-term memory, a swarm does not instantaneously adapt to the new conditions; instead, we observe a “wavefront” progressing through the snapshots shown in the left panels of Fig.

It has recently been suggested that changing boundary conditions in aeolian systems are associated with a temporary increase in the interaction density

While the simulations in the previous section all simulated a unimodal wind, many real-world swarms are exposed to a secondary mode for some parts of the year ^{−2} and

In Fig. ^{−2} and

Final states and stabilisation of mean size and dune number for ^{−2} and

Although

It was shown in

The distributions of asymmetry are shown in Fig.

Histograms of dune asymmetries defined at the ratio of port and starboard flank widths are shown in panels

A feature of barchan swarms that we have not yet discusses is that a downwind dune tends to align with the horn of its upwind neighbour

The normalised lateral offset of the nearest downwind neighbours (defined as the lateral offset divided by the width of the upwind dune) in the populations of

In Fig.

The lateral offset of the nearest downwind neighbour of dunes normalised by the width of the upwind dune for scaled outflux

The efficiency of the TFABM has meant that we have been able to perform a significant number of swarm-scale simulations and investigate the impact of a number of parameters on the types of swarms that are produced. Previous ABMs have not been able to produce the longitudinally homogeneous size distributions

The phenomenological parameter

Fragmentation collisions are able to reduce the size of a large dune through the creation of at least one additional new dune. However, these dunes are often too small to persists on their own, as has previously been reported using continuum simulations

If, on the other hand, the outflux from dunes scales with the influx they receive then, rather than an equilibrium size for dunes, there is an equilibrium flux for which a barchan of any size would be stable. In this case, longitudinal homogeneity can be achieved if collisions typically drive dunes to a particular size. Size selection has been shown to be possible using exchange collisions only

Looking at Fig.

In order to constrain the range of parameters in the TFABM to reproduce realistic swarms, the longitudinal size and density profiles of real-world swarms require further investigation. In Fig. 1 of

Assuming that swarms are longitudinally homogeneous, at least in size, the values of

On the other hand, we have shown in Fig.

Finally, we have been able to reproduce the two-peak distribution of the alignment of dunes and their downwind neighbours, which is something that has been observed previously for real-world swarms

We have demonstrated that the Two-Flank Agent-Based Model is capable of generating swarms of barchans which have longitudinally homogeneous size distributions which had not been produced using previous agent-based models but which are known to be the case for some real-world barchan swarms. The model and results presented here, therefore, represent a step forward in our understanding of the behaviour of these complex swarms. We observed that, depending on whether or not outflux of sand from dunes scales with the influx, different relative frequencies of collision types are required to produce longitudinal homogeneity. However, in both cases, the necessary collision dynamics coincide with those reported in numerical and experimental studies and are produced by the TFABM with

Longitudinal stability of dune density is a rarer occurrence in the simulated swarms; however, it is possible to create sparse swarms exhibiting this kind of homogeneity. Current quantitative evidence is lacking for the assessment of whether such stability is indeed a universal property of barchan swarms in nature. If it is a universal property, then it may be necessary to incorporate sand patch formation or additional sediment supply in the TFABM to produce dense homogeneous swarms.

Unlike with previous agent-based models, the Two-Flank Agent-Based Model is capable of simulating swarms under bimodal winds, which revealed that bimodality may be responsible for asymmetry distributions centred away from unity, and it also changes in the alignment of dunes with their neighbours. This adds to our understanding of how dune interaction shape the properties of swarms.

Being able to simulate more realistic swarms using an agent-based model presents many exciting opportunities for future work, including the investigation of changing boundary conditions. Preliminary results indicate that there is a finite transition time for barchan swarms, after which historical boundary conditions are no longer reflected in the properties of swarms. Future work could explore changing boundary conditions further to understand how swarms will be affected by anthropogenic change. It may also be possible to use the TFABM to train machine-learning algorithms to trace individual barchans through satellite imagery of dense swarms.

Two-Flank Agent-Based Model has been developed openly, and version 3 is published at

DTR led the development of the model, performed the simulations described, and wrote the initial and second draft of this paper. ACWB assisted in the design of the model and the research strategy and carried out text edits on the submitted paper.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We are grateful for the helpful suggestions of the reviewers which improved this work.

This research has been supported by the Engineering and Physical Sciences Research Council (grant no. EP/L015854/1).

This paper was edited by Tom Coulthard and reviewed by Dongxu Cai and one anonymous referee.