Shear flow dynamics described by the 2D incompressible Navier-Stokes equations is studied for a vorticity profile having two peaks, leading to Kelvin-Helmholtz instabilities at two resonant surfaces. The linear modes and the resulting nonlinear waves corresponding to the two surfaces have different phase velocities. The nonlinear behavior is studied as a function of two parameters, the Reynolds number and a parameter specifying the width of the peaks in the vorticity profile. For parameters such that the instabilities grow to a sufficient level, there is Lagrangian chaos, leading to mixing of vorticity, i.e. momentum transport, between the chains of vortices or cat's eyes. The Lagrangian chaos is measured by a "patchiness" diagnostic and by computing the distribution of finite time Lyapunov exponents. The momentum transport leads to interaction between the sets of vortices and a decrease in the relative phase velocity of the waves. For parameters for which the Lagrangian chaos covers a large enough region, and for sufficiently high Reynolds number, the Eulerian behavior of the flow develops chaos as measured by several time series. A discussion of the role of Lagrangian chaos and in what sense it is related to the observed Eulerian chaos is given.
In collaboration with Diego-Castillo-Negrette, ORAL.