Collective orientation of an immobile fish school and effect on rheotaxis

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.