LAUSR

We study the orientational order of an immobile fish school. Starting from the second Newton law, we show that the inertial dynamics of orientations is ruled by an Ornstein-Uhlenbeck process. This process describes the dynamics of alignment between neighboring fish in a shoal-a dynamics already used in the literature for mobile fish schools. First, in a fluid at rest, we calculate the global polarization (i.e., the mean orientation of the fish), which decreases rapidly as a function of noise. We show that the faster a fish is able to reorient itself the more the school can afford to reorder itself for important noise values. Second, in the presence of a stream, each fish tends to orient itself and swims against the flow: so-called rheotaxis. So, even in the presence of a flow, it results in an immobile fish school. By adding an individual rheotaxis effect to alignment interaction between fish, we show that in a noisy environment individual rheotaxis is enhanced by alignment interactions between fish.