Dynamical chaos in a minimum system of truncated Euler equations

Abstract

We derive a finite-dimensional Hamiltonian dynamical system, projecting the Euler equations onto a basis composed of six wave vectors closed into a tetrahedron with equal edges. We obtain a system of twelve equations for complex amplitudes of the flow, with three integrals of motion (IM's), namely, the energy $(E)$, helicity $(H)$, and an additional one related to the squared angular momentum. The system admits reduction to six complex equations and finally to six real ones for three positive-helicity and three negative-helicity modes, which is the minimum dynamical system to approximate the three-dimensional Euler equations. We simulate the latter system numerically and demonstrate that it is partly chaotic despite having the same three IM's in its six-dimensional phase space. The simulations reveal that the dynamics is fully chaotic in some region of the phase space, while in another region it is mixed: at the same set of values of IM's, some trajectories are chaotic and some are regular. A chart showing the fully chaotic and mixed regions is obtained, in the first approximation, in the space of the IM's values. To quantify the chaos, we compute the mean Lyapunov exponent (LE) $\ensuremath{\Lambda}$ characterizing the local instability of the trajectories. We find that $\ensuremath{\Lambda}$ is nearly independent of the choice of the trajectory at fixed values of IM's in the fully chaotic region, i.e., this region appears to be ergodic. Generally, the system is ``most chaotic'' at zero helicity and it is apt to become ``less chaotic'' with an increase of $H/E$. We demonstrate, in accord with this, that $\ensuremath{\Lambda}$ is a monotonically decreasing function of $H/E$ in the chaotic region, but inside the region of the mixed behavior the dependence is not monotonic. We also report some results obtained for a more general system of six complex equations. A preliminary inference is that there is no drastic qualitative difference from the system of six real equations, though a change (decrease) of LE can be conspicuous. A simple dissipative generalization of the model is considered too.