Steady motions of a continuous medium, resonances and Lagrangian turbulence

Abstract

A method which enables one to establish a non-regularity property of the motion of fluid particles (known as chaotic advection or Lagrangian turbulence) for typical steady flows is developed. The method is based on expanding solutions of the equations of motion of a continuous medium in powers of a small parameter and using the conditions for the destruction of invariant resonant tori when perturbations are added. It is shown that the velocity field, defined as the solution of the Burgers equations, generates a generally non-regular dynamical system. For an ideal barotropic fluid in an irrotational force field, the method proposed yields a well-known necessary condition for chaotization: the velocity field is collinear with its curl. Special attention is given to investigating the chaotization of typical steady flows of a heat-conducting perfect gas.