Hydrodynamics of Turning Flocks

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 wavelengths. This mode becomes unstable as a result of the growth of bend and splay deformations augmented by the spin wave, signaling the transition to complex spatiotemporal patterns of continuously turning and swirling flocks.

Figure 1

Flocking patterns obtained via numerical solutions of Eqs. (6)–(8) (top and middle rows) and via particle simulations using Eqs. (1)–(2) (bottom row). The left (right) column shows the patterns obtained at points $A$ $(B)$ of the phase diagram Fig. 2, corresponding to the spinodal portion of the coexistence region of traveling bands of polar liquid and disordered gas and to the region of the spin-wave instability, respectively. The snapshots (a)–(d) are obtained with $\tilde{\gamma }=2.0$, $\tilde{\chi }=0.5$ [(a),(c)] and $\tilde{\gamma }=7.6$, $\tilde{\chi }=1.7$ [(b),(d)] on a $300\times 300$ grid lattice with grid size 0.1, integration time step 0.002, and periodic boundary conditions. The particle simulations are performed with 3000 particles in a box of size $L=10$ with periodic boundary conditions. Simulation parameters are $R=1$, $\epsilon =2.0$, $v_0=2.0$, $\chi =1.0$, $\eta =1.0$, and $\gamma =0.16$ (e) or $\gamma =0.80$ (f). The integration time step is 0.01. In (a)–(d), the arrows represent the local polarization, with length proportional to the polarization strength. The color indicates spin-current density in (a)–(b) and number density in (c)–(d). In (e)–(f), the arrows represent polarization of individual particles (see movies in Supplemental Material [24]).

Figure 2

(a) Phase diagram in the plane of dimensionless $\gamma$ and $\chi$. (b) Phase diagram in the plane of dimensionless $\gamma$ and $\theta$ at $\chi =1.0$. The shaded region $A$ is the spinodal portion of the region of coexistence of disordered gas (existing for $\gamma <2$) and traveling bands of polar liquid (existing for $\gamma >8/3$). The coexistence region is delimited by the binodals (not shown) and extends inside the white regions, both to the right and to the left of region $A$, as verified via particle simulations. The shaded region $B$ corresponds to the region where the homogeneous polar liquid is linearly unstable to spin waves, as shown in Fig. 1. (c) Real part of the dispersion relation of the transverse mode ${\sigma }_t^{\pm }={\sigma }^{\pm }(\pi /2)$ at $\chi =1$ and $\gamma =7.0$, 8.0, 9.0, and 10.0. (d) Real part of the dispersion relation of the transverse mode ${\sigma }_t^{\pm }={\sigma }^{\pm }(\pi /2)$ at $\gamma =9$ and $\chi =0.5$, 1.0, 1.5, and 2.0.

Figure 3

(a) Snapshot of the anisotropic spin wave in the polarized state at $\gamma =7.0$ and $\chi =2.0$. Color indicates the spin-current density. (b) Speed of spin waves in the polarized state as a function of alignment strength $\gamma$ for $\chi =1.0$, 1.5, 2.0 (red, blue, black) in the directions longitudinal (circles) and transverse (squares) to that of mean polarization. The dashed line is the transverse speed $|c^{\pm }(\pi /2)|$ in Eq. (10). The system is evolved for 5000 time steps.