Lyapunov exponents and the dimension of periodic incompressible Navier—Stokes flows: numerical measurements

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Abstract
In this paper we study the quasi-stationary turbulent state developed by an incompressible flow submitted to a constant periodic force. The turbulent state is described by its Lyapunov exponents and Lyapunov dimension. Our aim is to investigate in particular if the dimension is, as several lines of argument seem to indicate, given in one way or another by the number of ‘turbulent’ modes present in the flow.The exponents may be viewed as the asymptotic limits of local exponents, which are the divergence rates between the actual state of the fluid and nearby states. These local exponents fluctuate as the fluid suffers chaotic changes: they are systematically larger during bursts of excitation at large scale. In one case, we find that the flow oscillates repeatedly, on a very long timescale, between two distinct turbulent states, which have distinct local Lyapunov spectra and also distinct energy spectra. At the moderate resolutions studied here (642, 1282 and 163), we find that the dimension of the attractor in flows with standard dissipation is bounded above by the number of degrees of freedom contained in the large scales, i.e. at wavenumbers smaller than that of the injection wavenumber kf. We also consider two-dimensional flows with artificial terms (hyperviscosity), which concentrate small-scale dissipation in a narrow band of wavenumbers, and allow the formation of an inertial range. The dimension in these flows is no longer bounded by the number of large scales; it grows with, but remains significantly smaller than, the number of modes contained in the inertial range. It is conjectured that this is due to the presence of coherent structures in the flow, and that at higher resolutions also, the presence of coherent structures in turbulent flows will lead to an effective attractor dimension significantly lower than the estimations based on self-similar Kolmogorov theories.