Abstract
This thesis concerns modelling the many-body behaviours of barchan dunes, a class of aeolian bedform which display a distinctive crescentic shape and migrate rapidly. Such dunes form in areas where there is only limited variability in the wind direction and the overall supply of sand is relatively low. Under these conditions, the available sediment forms into barchan dunes which migrate at a size-dependent rate in the direction of the dominant wind and are separated by bare non-erodible ground. While these dunes can occur as isolated bedforms or in small numbers, they are more typically found in large systems which we refer to as swarms; the largest of which span many hundreds of square kilometres and contain tens of thousands of dunes. Within these swarms, size-dependent migration rates ensure that collisions are frequent occurrences. Although fast-moving in geological terms, with typical migration rates of the order of tens to hundreds of metres per year, the timescale over which a collision is completed has ensured that field observations of collisions of these bedforms are only limited. In recent decades, the availability of high spatial resolution aerial and satellite photography has improved this situation, however, the length of time that such images have been available together with the limited time resolution of these images means that the number of collisions which have been identified in this manner is insufficient for a full exploration of the phase space of real-world barchan collisions. However, the development of continuum models and cellular automata simulating wind-blown sand has enabled the study of the entire process of collisions over a much greater range of initial conditions. These simulations have been complemented by laboratory experiments in which proxies of aeolian dunes are created using glass beads in a dense medium such as water for which both the time and length scales are reduced, once more enabling entire collisional processes to be analysed. From these studies, it has been discovered that collisions can result in aggregation (or merging), material exchange, or fragmentation into multiple dunes. Aside from collisions, many studies have suggested that barchans can fragment without a collision taking place either due to variations in the wind or the presence of upwind bedforms; a process known as calving. In extreme winds, the increased flux streaming off barchans has also been observed to coalesce into new bedforms in the lee of the existing dunes. Even without such extreme conditions, downwind dunes may still absorb sand flux emitted by upwind neighbours, a form of long-range interaction.The abundance of both collisional and long-range interactions between dunes means that swarms must be treated as interacting many-body systems rather than collections of isolated bedforms. Under this context, the research questions which formed the core aspects of my PhD were:
1. What are the properties of barchans in swarms which cannot be explained from an understanding of
isolated dunes?
2. How do the interactions between dunes produce these observed properties?
3. What effect does a change in the nature of interactions have on the observed properties?
To answer these questions I undertook three major projects as part of the PhD, each of which will form a section of this thesis following on from an initial section in which I review the existing literature on barchan dunes. The major projects of my PhD were:
1. Mean-field modelling of barchan swarms.
2. Analytical modelling of asymmetry growth.
3. Agent-based modelling of barchan swarms.
The first project culminated in two papers published in peer-reviewed journals. For the second project, I produced a letter which is currently under review for publication. The final project also yielded two papers, one of which has been published and the other of which is under review. For all of these research articles, I was the lead author. I have also presented my work at several international conferences and workshops.
In mean-field models, one simplifies the system into an abstract set of objects, foregoing any consideration of the space they occupy. Typically, this type of model can only be applied to systems which are at least three dimensional whereas barchan swarms are two dimensional systems. However, the restriction of high dimensionality is imposed on mean-field models to ensure that the systems are sufficiently well-mixed so that every object is able to interact with every other object. The size distributions of barchan swarms have, however, been shown to be spatially homogeneous which means that, if one considers a region in the centre of a large swarm, the dunes entering and leaving that region should have the same distribution of sizes. This ensures that the assumption of sufficient mixing can be achieved despite the low dimensionality of the systems. Two existing studies have made use of mean-field models to study barchan swarms. The first of these featured an incomplete set of collision phenomena and no spontaneous fragmentation. The second mean-field model included a more complete set of collision types but did not include spontaneous fragmentation and included other features which are not observed in real-world swarms. To account for the limitations of the existing mean-field models of barchan swarms, the first major project of my PhD was the development of a novel general mean-field model which could include any process which conserves a single quantity such as volume/mass in the case of barchans. The specific aims of this work were to
1. Create a mean-field model capable of including all known interactions present in barchan swarms.
2. Understand how changing the types and forms of interaction shape the observed distributions.
The model was developed from two classes of models, aggregation-fragmentation models which have been applied to many physical and chemical systems, and asset-exchange models which have been developed to model economic systems. In existing aggregation-fragmentation models, one usually considers the objects to be polymers, that is, collections of a discrete number of basic building blocks or monomers. When two objects collide in such an aggregation-fragmentation model, the total number of monomers remains constant, but the polymers tend to either coalesce into a single large polymer or fragment into a collection of monomers. While other interactions, such as spontaneous fragmentation, have also been included, the vast majority of aggregation-fragmentation models consider discretised objects, a perspective which is not valid for the study of barchan dunes.
Asset-exchange models, on the other hand, which have been shown to recreate properties of wealth distributions and class formation in socio-economic systems, one typically considers the conserved quantity e.g. wealth, to be a continuous quantity which is more appropriate for application to barchans for which we can simply replace wealth with mass/volume. However, as the name suggests, most asset-exchange models consider only exchange interactions, and very few have considered merging or fragmentation-type interactions.
Based on the continuous nature of asset-exchange models and the wider range of interaction phenomena allowed in aggregation-fragmentation models, I developed a new mean-field model which allows for any process with n inputs and m outputs in which a single, continuous quantity is conserved. Allowing for n-body interactions ensures that the model is capable of studying all possible conservative interactions. Multi-body interactions may be possible in barchan swarms due to the timescale associated with collisions and are possible in economic systems for instance in multiple mergers.
Despite the generality of the model, I derive analytical expressions for all of the integer moments for the resulting steady-state distribution of the conserved quantity. Furthermore, I show that one can express the solution for the entire steady-state probability distribution in a form which is compatible with an existing algorithm, allowing the steady-state distributions to be calculated without having to perform a full set of numerical simulations. In the case of two-body collisions only, I also show that it is possible to calculate not only the steady-state moments but also the full time evolution of the moments and the size of fluctuations in finite systems.
The importance of this work is that it demonstrates that, even in a complicated system with generalised interactions, the properties of the steady-state distribution can be explicitly related to the rules governing the interactions. Additionally, the model allows me to explain the distributions observed from one of the previous mean-field models of barchans. This theoretical work was published in the Journal of Statistical Mechanics: Theory and Experiment (JSTAT). The applicability of the work in the study of multiple systems has led to its being cited by studies in fields including Earth science, applied mathematics, and economics.
My objective in creating the mean-field model was to make it as general as possible so that it could be applied to many different types of systems. However, it was then important to tailor the specifics of the models to be applicable to barchan swarms. Before applying the model to such systems, it was important to have data from real-world swarms to which the model outputs could be compared. To do this, I looked at satellite images from Earth and Mars and drew areas for the zones to be studied. I then manually recorded on each dune within these marked areas the positions of seven points: the toe, leftmost extent, tip of the left horn, centre of slipface base, crest, tip of the right horn, and rightmost extent. For bedforms without a visible slipface, I recorded only the upwind, downwind, leftmost, and rightmost extents. Finally, for complex/deformed bedforms, I traced around the area. Altogether, dunes in six zones were measured; three in Tarfaya, Morocco, one in Mauritania, and two in the northern circumpolar region of Mars. The total number of bedforms I recorded was 6392 of which 5825 (91%) were identified as barchans. From the positions of these points around the base of the barchans, many morphological characteristics of the dunes can be calculated.
Since the mean-field model is used for calculating the distribution of a conserved quantity, it was necessary to calculate the volume of the dunes. Fortunately, it is well-established that the volume of barchans is proportional to the cubic power of its linear dimensions (width, length, height). The easiest of those dimensions to extract from the dunes was the length, and so we used the lengths of the dunes to estimate their volumes, giving volume distributions for each of the six zones of study. We found that the barchans in each of the terrestrial zones had a mean length of 23m while the Martian dunes were much larger with mean lengths of 113m and 131m. Since the model concerns how interactions within a system govern its size distribution, the overall scale is not as important as the shape of the distribution. Therefore, all measurements were normalised such that the dimensionless mean volume of dunes in the swarms was unity. When this was done, we found a striking similarity between the Mauritanian and Martian swarms which appeared to have almost identical volume distributions. The three Tarfaya swarms had very similar distributions to one another but with a shape that was distinct from the Mars/Mauritania distribution.
In order to understand the difference between the distributions in Tarfaya and Mars/Mauritania, I compared the observed distributions to the outputs of the mean-field model. Rather than consider the fully general case I instead implemented two specific cases one in which I allowed for exchange collisions only but did not fix the rule for the output sizes during a collision, and one where I allowed for Calving, Aggregation, Fragmentation, and Exchange (the CAFE model) where we constrained, where possible, the interactions rules to match with empirical observations from continuum simulations of barchan collisions. I found values for the free model parameters in each model implementation by comparing the integer moments of the observed distributions to the analytical expressions we had derived.
The results reveal that, using the empirical collision rule, the narrow Mars/Mauritania distribution is best fit when most interactions were exchange collisions while the Tarfaya distribution is more closely reproduced when aggregation and fragmentation collisions dominated. In both cases, calving events were rare. On the other hand, the exchange-only model demonstrates that the distributions can both be produced using only exchange collisions but where the collision rules are very different between the Tarfaya and Mars/Mauritania distributions. These findings suggest that the size distributions in real-world swarms are intrinsically linked to the frequency and output distributions of the different types of interactions. However, the fact that both the CAFE model and exchange-only model are able to reproduce the observed distributions suggests that more information is needed to understand whether it is the frequency of collisions or the form of the output that leads to the varied distributions observed in swarms in nature. This work was published in Physica A: Statistical Mechanics and its Applications.
With the data I collected, we have access to many other properties of swarms such as spatial properties which cannot be accessed using a mean-field model, implying that a more complex type of model, agent-based modelling, is required. Before moving onto agent-based modelling, however, in the second major project of my PhD, I explored some other properties of the dunes I had measured, specifically the asymmetry of the dunes.
Barchans in nature are rarely the perfectly symmetric shape that is described in idealised models. Considering a definition of bilateral asymmetry of the dunes by dividing them into a port and starboard flank, the divisions between the two flanks being the central axis of the dune along which fall the toe, crest and slipface apex. We then defined an asymmetry ratio as the ratio of the widths of the larger and smaller flanks. We observed that this ratio was typically smaller for larger dunes and wanted to explain this size dependence. To this end, we developed a simple model which considered each of the two flanks to grow independently of the other. In the first instance, the rate at which the flanks changed size is expressed as a general polynomial of flank size. In this general case, the equation of motion of the asymmetry ratio does not have an analytical solution. However, I show that when one of the terms in the polynomial dominates, an analytical expression for the size dependence of bilateral asymmetry can be derived.
We compare the observed size dependence with the solutions to the equations of motion when the different terms are assumed to dominate. From this, I reveal that the model where dune flanks grow at a rate proportional to their area gives the best fit for all six of the zones we measured. This contrasts with the typical view that the rate at which a barchan grows should be a linear function of its width. We do find, for the Martian and Mauritanian zones, that a rate of growth proportional to the flank width gives almost as good of a fit as the area-dominated model, suggesting that in those regions there may be an interplay between the two modes of growth whereas in Tarfaya area-dominated growth appears the most likely. We propose that area-dependent evolution may arise due to variability in the wind direction, the timescale of internal restructuring processes, or due to dune collisions. Although these explanations are tentative and further work is needed to understand the link between size and asymmetry, the closeness of the fit of the area-dominated model is still an important finding. Furthermore, the finding that the Mauritanian and Martian dunes show similar trends in the size-dependence of bilateral asymmetry suggests once more that the interactions governing those swarms are different to those governing the behaviour in Tarfaya as was the case for the size distributions themselves.
Having explored several aspects of the datasets I had collected using simple analytical models, it was then time to create a more realistic model of barchan swarms; an agent-based model and the third major project of my PhD. Based upon the success we had had with the simple two-flank model of asymmetry growth, I adpoted the same two-flank view of the dunes in our agent-based model, naming it the Two-Flank Agent-Based model (TFABM). However, unlike before when we were constrained by limiting ourselves to behaviour which was analytically tractable, moving to an agent-based model allowed for the inclusion of more complex and realistic phenomena, chief among which was the inclusion of a process which couples the growth of the flanks of a dune such that they are no longer fully independent. This lateral flux is likely to combine asymmetries in the motion of sediment during periods of oblique winds together with internal restructuring processes, avalanching, and reptation on the dune itself. The rate at which this shifting of material occurs between the flanks is assumed to be proportional to the difference between the lengths of the flanks.
When the rate of lateral flux exceeds that of the longitudinal flux, the dunes are no longer behaving in the manner of a typical barchan and thus, we argue, that dunes which reach that level of asymmetry will calve (spontaneously fragment) with each of the flanks breaking off and forming into separate symmetric barchans. The same fragmentation condition is also used to determine the outputs of collisions which occur when the centre of mass of a flank of one dune intersects with another dune. When a collision occurs, all of the intersecting flanks merge together while the others merge provided they are similar enough in size to not meet the calving threshold. By using the same rule for calving and collisions, the model’s parameter space is reduced.
The collision rule is able to reproduce the majority of collisions observed in microscopic models while maintaining the efficiency of agent-based modelling. Furthermore, we show that not only is the model able to reproduce collision behaviour and calving, but since we use a much more realistic shape than previous agent-based models, we are able to use the model in instances of variable wind. We demonstrate that the model is capable of reproducing the growth of asymmetry under the bimodal winds. When the angular separation between the modes is an acute angle we find that asymmetry grows according to the so-called, Bagnold model while for obtuse angular separation, the growth was described by the Tsoar model. Finally, an asymmetric dune subject to a unimodal wind reverts to a symmetric dune. The model structure and description of the phenomena it captures were described in a paper published in Geophysical Research Letters (GRL). Once we had established that the TFABM was able to reproduce the phenomenology of real-world barchans, the next task was to apply the model in simulations of barchan swarms. The questions which we sought to address with this work were:
1. How does changing the rate of lateral flux affect the observed swarms?
2. How do the boundary conditions influence the swarms’ properties?
3. How does a steady-state system respond to a change in boundary conditions?
4. How does wind variability change the properties of swarms?
The lateral flux term is not only the main process which drives dunes away from an asymmetric morphology but also the main model parameter which affects rates of calving and the rate and output sizes of collisions. Thus, we show that changing the lateral flux parameter significantly alters the swarms that are produced using the model, with values of the parameter leading to denser swarms with smaller average dune size. This can be explained by noticing that as the parameter is increased, the typical outcome of collisions transitioned from initially aggregation/exchange dominated, to fragmentation dominated. This transition also coincides with a transition to swarms which exhibit longitudinal homogeneity of size distribution which is observed in real-world swarms. Interestingly, we find that calving is infrequent compared to collisions, as we find from mean-field modelling.
Varying the lateral flux parameter also affects the spread of the asymmetry ratio distributions, with lower values of the parameter leading to wider distributions. Comparing the distributions in the simulated swarms to those that we observed in the measured swarms, reveals that the range of values at which stabilised longitudinal size distributions begin to occur is the same range of values that reproduce asymmetry distributions with the same standard deviation as the natural swarms.
Aside from the lateral flux parameter, we find that a major control on the properties of the simulated swarms is the rate at which dunes are injected into the system at the upwind boundary. We find that increasing the rate of injection leads to dunes with higher number density and smaller dunes. However, for higher density swarms we find that the number density decreases with downwind distance unlike in realworld swarms.
When a swarm is allowed to stabilise under particular boundary conditions which are then changed, we observe that the new steady-state properties pass through the swarm in the manner of a wave. However, we do not observe a higher rate of collisions during this transition as has been suggested in some works.
Finally, under bimodal winds, we observe that overall dunes move away from their usual symmetric shape. A property of spatial structuring observed in both the real-world swarm and those simulated with a unimodal wind is that the downwind neighbour of an upwind dune typically aligns such that its toe is immediately downwind of the horn of the upwind dune. Under bimodal winds with large angular separation from the primary mode, this alignment at first becomes skewed to favour one horn rather than the other, and then entirely vanishes. Due to the preferential alignment with a particular horn under bimodal winds, in contrast to the case of isolated dunes, we do not observe asymmetry growth according to the Tsoar model even when the angular separation of the wind modes is obtuse.
The different projects I undertook during the thesis each contributed to my understanding of the behaviours of barchan swarms in important ways. The mean-field modelling was useful in revealing how both the rates of the different types of interactions and the form that those interactions take, impact the size distribution of such swarms. I was able to explore these links between barchan-scale processes and swarm-scale properties further in my analytical modelling of asymmetry growth. While simplistic in their form, both the mean-field model and analytical asymmetry growth models allowed me to explore many facets of barchan swarms and gave invaluable insights into how these systems evolve. Without these insights, the development of the Two-Flank Agent-Based model would not have been possible. The model shares similarities with my mean-field modelling, most notably, a parameter allowing me to control both the rates and forms of barchan collisions and calving. Furthermore, the dunes in the model are very similar in structure to those considered in the analytical asymmetry growth modelling.
The study of the form of barchan interactions is ongoing with several research groups currently exploring the phase space of these interactions. My work represents an important bridge linking the findings of such studies with observable properties of real-world swarms. Such a link is vital if we are to be able to understand how swarms will behave in the future particularly given uncertainty inherent in a changing climate. Furthermore, machine learning methods to automatically detect and track barchans are currently hampered by the poor time resolution of available imagery. The number density of barchans in swarms and their fast migration means that, without higher time resolution, machine learning techniques struggle to track individual bedforms between the available images. A realistic agent-based model offers a solution to this problem since it should be able to predict the future positions and sizes of dunes given an initial state and therefore can be used to train the next generation of neural networks applied to study these systems.
Altogether, the key findings of the thesis are:
1. The interactions at play in barchan swarms are the major factor determining the size distribution and asymmetry distributions.
2. Swarms with similar size distributions also display similar asymmetry distributions.
3. Asymmetry is a size-dependent phenomenon influenced by the rate at which dunes grow.
4. External drivers such as boundary conditions and wind variability have more of an influence on spatial properties such as the number density and alignment of neighbouring dunes.
5. The transfer of material through an asymmetric dune is an important control of collision dynamics and calving.
6. A rate of this lateral transfer of material which is consistent with fragmentation conditions dominating is also consistent with producing longitudinally homogeneous size distributions and replicating the width of observed asymmetry distributions.
| Date of Award | 1 Jun 2024 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Andreas Baas (Supervisor) & Alessia Annibale (Supervisor) |