STOCHASTIC DYNAMICS OF VISCOELASTIC SKEINS: CONDENSATION WAVES AND CONTINUUM LIMITS

Abstract

Skeins (one-dimensional queues) of migrating birds show typical fluctuations in swarm length and frequent events of "condensation waves" starting at the leading bird and traveling backward within the moving skein, similar to queuing traffic waves in car files but more smooth. These dynamical phenomena can be fairly reproduced by stochastic ordinary differential equations for a "multi-particle" system including the individual tendency of birds to attain a preferred speed as well as mutual interaction "forces" between neighbors, induced by distance-dependent attraction or repulsion as well as adjustment of velocities. Such a one-dimensional system constitutes a so-called "stochastic viscoelastic skein." For the simple case of nearest neighbor interactions we define the density between individualsu = u(t, x) as a step function inversely proportional to the neighbor distance, and the velocity function v = v(t, x) as a standard piecewise linear interpolation between individual velocities. Then, in the limit of infinitely many birds in a skein of finite length, with mean neighbor distance δ converging to zero and after a suitable scaling, we obtain continuum mass and force balance equations that constitute generalized nonlinear compressible Navier–Stokes equations. The resulting density-dependent stress functions and viscosity coefficients are directly derived from the parameter functions in the original model. We investigate two different sources of additive noise in the force balance equations: (1) independent stochastic accelerations of each bird and (2) exogenous stochastic noise arising from pressure perturbations in the interspace between them. Proper scaling of these noise terms leads, under suitable modeling assumptions, to their maintenance in the continuum limit δ → 0, where they appear as (1) uncorrelated spatiotemporal Gaussian noise or, respectively, (2) certain spatially correlated stochastic integrals. In both cases some a priori estimates are given which support convergence to the resulting stochastic Navier–Stokes system. Natural conditions at the moving swarm boundaries (along characteristics of the hyperbolic system) appear as singularly perturbed zero-tension Neumann conditions for the velocity function v. Numerical solutions of this free boundary value problem are compared to multi-particle simulations of the original discrete system. By analyzing its linearization around the constant swarm state, we can characterize several properties of swarm dynamics. In particular, we compute approximating values for the averaged speed and length of typical condensation waves.