When and why does bioturbation lead to diffusive mixing?

Bioturbation in aquatic sediments results from many different biological activities, inducing particle displacement over a variety of length and time scales. Despite this inherent complexity, empirical tracer studies show that bioturbational mixing is often well described by a simple diffusive model. To resolve this apparent contradiction between biological complexity and modeling simplicity, we present an investigation into the diffusive nature of bioturbation. To this end, we examine a stochastic description of bioturbation, where particle mixing is described as a sequence of random bioturbation events. Particle movement is governed by three basic variables: the direction of jumping, the jumping distance, and the waiting time between two bioturbation events. A set of gradually more complex random-walk models is constructed by treating these variables consecutively as stochastic. Each time, the conditions are determined under which these random walks generate diffusive behavior. One important criterion is that the tracer should experience a sufficient number of bioturbation events, and so the time scale of observation should be large enough. The full list of conditions needed to arrive at diffusive mixing is different and more extensive than in past examinations of the topic. This analysis clarifies the relation between the biodiffusion model and more sophisticated mixing models, revealing that the conventional distinction between local and nonlocal models is not entirely relevant. Instead, we propose a distinction between normal or anomalous mixing, which is based on the intrinsic nature of the faunal activities causing particle dispersal. Overall, our analysis provides insight into the conditions under which the biodiffusion model can be applied to tracer profiles, and also shows why it sometimes fails, as is the case of sediments dominated by head-down deposit feeders.