Topology of trajectories of the 2D Navier-Stokes equations

In spectral form the 2D incompressible Navier-Stokes equations in a square periodic region will be represented by 430 complex Fourier amplitudes which correspond to isotropic truncation of the upper wave number 16. For small viscosity, we have found five equilibrium states I-V in the entire range of forcing; I-fixed point, II-circle, III-closed orbit, IV-torus, and V-chaos. The fixed-point equilibrium state is the laminar flow. As the forcing passes through a critical value, the fixed point evolves directly to equilibrium state III under a typical multimode forcing. The chaotic transition takes place on a 2-torus-like manifold (equilibrium state IV) which is the product space of a circle and the closed orbit of equilibrium state III, similar to the quasiperiodic 2-torus of Ruelle and Takens. For sufficiently large forcing, the evolution of equilibrium state V is nothing but a simulation of quasistationary 2D turbulence. From the Lyapunov exponents of turbulent flows, we have evaluated the constants in the theoretical results of Foias and his colleagues, which relate the determining mode and fractal dimension with the enstrophy dissipation wave number of 2D turbulence.