Optimal trajectories for Bayesian olfactory search in turbulent flows: The low information limit and beyond

In turbulent flows, tracking the source of a passive scalar cue requires exploiting the limited information that can be gleaned from rare, randomized encounters with the cue. When crafting a search policy, the most challenging and important decision is what to do in the absence of an encounter. In this work, we perform high-fidelity direct numerical simulations of a turbulent flow with a stationary source of tracer particles, and obtain quasi-optimal policies (in the sense of minimal average search time) with respect to the empirical encounter statistics [1-3]. We study the trajectories under such policies and compare the results to those of the infotaxis heuristic. In the presence of a strong mean wind, the optimal motion in the absence of an encounter is zigzagging (akin to the well-known insect behavior "casting") followed by a return to the starting location. The zigzag motion generates characteristic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msup><mml:mi>t</mml:mi> <mml:mrow><mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:msup> </mml:mrow> </mml:math> scaling of the rms displacement envelope. By passing to the limit where the probability of detection vanishes, we connect these results to the classical linear search problem and derive an estimate of the tail of the arrival time pdf as a stretched exponential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>∼</mml:mo> <mml:mtext>exp</mml:mtext> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>k</mml:mi> <mml:msqrt><mml:mi>T</mml:mi></mml:msqrt> <mml:mo>)</mml:mo></mml:mrow> </mml:math> for some <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn></mml:mrow> </mml:math> , in agreement with Monte Carlo results. We also discuss what happens as the wind speed becomes smaller.