  We present a hydrodynamic model of flocking that generalizes the familiar
Toner-Tu equations to incorporate turning inertia of well-polarized flocks. The
continuum equations controlled by only two dimensionless parameters,
orientational inertia and alignment strength, are derived by coarse graining
the inertial spin model recently proposed by Cavagna et al. The interplay
between orientational inertia and bend elasticity of the flock yields
anisotropic spin waves that mediate the propagation of turning information
throughout the flock. The coupling between spin current density to the local
vorticity field through a nonlinear friction gives rise to a hydrodynamic mode
with angular-dependent propagation speed at long wavelength. This mode goes
unstable as a result of the growth of bend and splay deformations augmented by
the spin wave, signaling the transition to complex spatio-temporal patterns of
continuously turning and swirling flocks.

Soft Condensed Matter;Statistical Mechanics;Biological Physics;Fluid Dynamics