A continuum model of bioconvection with a centripetal force

Abstract

The paper is devoted to mathematical modeling of the unusual collective behavior of emperor penguins in winter. Recent observations of flocks of penguins of this species in their natural habitat have shown that penguins use various forms of adaptive behavior that help them survive in the conditions of the Antarctic winter. Despite the fact that each penguin is well protected from cold, these sea-birds are periodically forced to gather in dense groups, within which the temperature necessary for survival is maintained. Observations have revealed the effect of a sudden liquefaction of a dense huddle of penguins, accompanied by a macroscopic circulation of birds from the edge of the group to its center and back. This paper is the first to assert that the observed phenomenon is bioconvection, which should be interpreted as a second-order phase transition. Moreover, the circulation of birds in a dense group can be explained by the mechanism of thermal convection, with the desire of penguins to move toward the temperature maximum playing the role of an effective force field. In the continuum approximation, we propose a mathematical model of the phenomenon, which is reduced to the equation of a self-gravitating porous disk saturated with an incompressible fluid that generates heat. We derive the governing equations in the Darcy-Boussinesq approximation and formulate a nonlinear boundary value problem. An exact solution of the linearized problem for infinitesimal perturbations of the base state is obtained, and the critical values of the control parameter for the onset of bioconvection are calculated. For finite-amplitude perturbations, the boundary value problem is solved using a finite-difference method. We show that in the case of axial symmetry, the most dangerous is the four-vortex motion. In the case of wind breaking the symmetry of the problem, the two-vortex circulation of penguins becomes the most dangerous. The numerical results obtained based on our theoretical developments are compared with observations of penguins in the Antarctic.