<p>It is well known that barchan dunes are not isolated bedforms but are able to interact with one another both directly, through collisions and the emission/absorption of flux, and indirectly, due to the effects of turbulence in the wake of a dune.&#160; In recent years, wave-tank experiments, continuum simulations, and cellular automata models have enabled researchers to model barchan-barchan interactions.&#160; The findings from these studies have been fed into object-based models of entire fields of barchans and used to predict the size distributions.&#160; Although there has been some success with these techniques, each model has failed to reproduce certain known properties on the field-scale; for instance, that the mean width is constant with downwind distance.&#160; Furthermore, previous attempts have not been based on a theoretical understanding of the role of interactions in determining the dune size distribution, thus limiting their potential as universal models of barchan swarms.</p><p>Mean-field models are relatively simple in terms of the mathematics, but have shown some degree of success in the modelling of barchan fields, although previous work has&#160; focused only on specific cases of interaction rules.&#160; We have developed a more general mean-field model which can include many different forms of interaction, making it applicable to a variety of problems, including socio-economic systems as well as fields of interacting barchans.&#160; Despite the generality of our model, we have been able to derive expressions for the dependence of the steady-state size distribution, and its moments, on the choice of interaction rules.&#160; This means that, by making a measurement of the size distribution of a barchan field, we are able to infer properties of the interactions at play.&#160;</p><p>To demonstrate the power of such a model we have measured size distributions of several barchan fields in the area of Tarfaya, Morocco.&#160; Measurements were made by recording locations of seven distinct points on each barchan to yield morphometric parameters of each dune and compile the size-distribution.&#160; By comparing the distribution and its moments to those predicted by the model, we can infer certain properties of the interaction rules, such as the relative probabilities of the different forms of collision.&#160; The results show an example of how our model provides a more comprehensive understanding of the way in which dune-dune interactions determine properties on the scale of the field.&#160;</p>
Effect of reagent type on the froth floatation of Sokoto phosphate ore