  We study the efficiency of search processes based on Levy flights (LFs) with
power-law distributed jump lengths in the presence of an external drift. While
LFs turn out to be efficient search processes when relative to the starting
point the target is upstream, in the downstream scenario regular Brownian
motion turns out to be advantageous. This is caused by the occurrence of
leapovers of LFs, due to which LFs typically overshoot a point in space. We
establish criteria when the combination of the external stream and the initial
distance between the starting point and the target favors LFs over regular
Brownian search. Contrary to the common belief that LFs with a stable index
alpha=1 are optimal, we find that the optimal alpha may range in the entire
interval (1,2) and even include Brownian motion as the overall most efficient
search strategy.
