Supergranulation as an Emergent Length Scale

Abstract

We have developed an n-body diffusion-limited aggregation model to simulate the dispersal and interaction of small-scale magnetic elements at the solar surface. The model is highly simplified and is based on the observation that small-scale magnetic elements are passively advected by the granular flow, which we approximate as a random walk. With a great many magnetic elements executing this random walk simultaneously, collisions are inevitable. We assume that these collisions lead to the aggregation of the colliding magnetic elements (if they have the same polarity) or mutual cancellation (if opposite polarity). In a similar fashion, the resulting clusters can subsequently interact with more magnetic elements or with other clusters. The clusters also undergo a random walk. However, the step size is reduced and the lifetime is increased in order to mimic the observation that larger magnetic flux concentrations move slower and live longer than smaller ones. The essential finding is that this process can produce a spatial distribution of clusters comparable to the supergranule cell pattern (depending on model parameters). The characteristic length scale associated with the spatial distribution of the clusters is quite sensitively dependent on the injection rate of fresh magnetic elements—when the injection rate is high (low) the length scale is small (large). This property provides a natural explanation for the observation that supergranule cells tend to be smaller when and where the level of magnetic activity is higher. We also find that at length scales similar to supergranulation the dominance of a given polarity tends to be enhanced, in comparison to the case where the same clusters are situated randomly in space. This is potentially testable by observation.