Intermittency in nonlinear dynamics and singularities at complex times

Abstract

High-pass filtering of turbulent velocity signals is known to produce intermittent bursts. This is, as shown, a general property of dynamical systems governed by nonlinear equations with band-limited random forces or intrinsic stochasticity. It is shown that singularities for complex times determine the very-high-frequency behavior of the solution and show up in the high-pass filtered signal as bursts centered at the real part of the singularity and with overall amplitude decreasing exponentially with the imaginary part. Near a singularity, nonlinear interactions, however weak they may be on the real axis, acquire unbounded strength. Investigations of singularities by nonperturbative methods is thus essential for quantitative analysis of high-frequency or high-wave-number properties. In contrast to results based on two-point closures, the high-frequency dissipation-range spectrum is actually not universal with respect to the low-frequency forcing. Unlimited intermittency is demonstrated, i.e., the flatness of the high-pass filtered solution grows indefinitely with filter frequency. This gives strong support to a conjecture of Kraichnan [Phys. Fluids 10, 2080 (1967)] about intermittency in the dissipation range of turbulent flows. The analysis is carried out in great detail for the nonlinear Langevin equation $m\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{v}=\ensuremath{-}\ensuremath{\gamma}v\ensuremath{-}{v}^{3}+f(t)$. Lorenz's three mode system and Burgers's model are also discussed. Conjectures are made about Navier-Stokes turbulence which can be checked experimentally and numerically.