  In this paper, we study the macroscopic limit of a new model of collective
displacement. The model, called PTWA, is a combination of the Vicsek alignment
model and the Persistent Turning Walker (PTW) model of motion by curvature
control. The PTW model was designed to fit measured trajectories of individual
fish. The PTWA model (Persistent Turning Walker with Alignment) describes the
displacements of agents which modify their curvature in order to align with
their neighbors. The derivation of its macroscopic limit uses the non-classical
notion of generalized collisional invariant. The macroscopic limit of the PTWA
model involves two physical quantities, the density and the mean velocity of
individuals. It is a system of hyperbolic type but is non-conservative due to a
geometric constraint on the velocity. This system has the same form as the
macroscopic limit of the Vicsek model (the 'Vicsek hydrodynamics') but for the
expression of the model coefficients. The numerical computations show that the
numerical values of the coefficients are very close. The 'Vicsek Hydrodynamic
model' appears in this way as a more generic macroscopic model of swarming
behavior as originally anticipated.
