Chaos and predictability of homogeneous-isotropic turbulence

We study the chaoticity and the predictability of a turbulent flow on the basis of high-resolution direct numerical simulations at different Reynolds numbers. We find that the Lyapunov exponent of turbulence, which measures the exponential separation of two initially close solution of the Navier-Stokes equations, grows with the Reynolds number of the flow, with an anomalous scaling exponent, larger the one obtained on dimensional grounds. For large perturbations, the error is transferred to larger, slower scales where it grows algebraically generating an " inverse cascade " of perturbations in the inertial range. In this regime our simulations confirm the classical predictions based on closure models of turbulence. We show how to link chaoticity and predictability of a turbulent flow in terms of a finite size extension of the Lyapunov exponent. The strong chaoticity of turbulence does not spoil completely its predictability. Such apparent paradox is related to the hierarchy of timescales in the dynamics of turbulence which ranges from the fastest Kolmogorov time to the slowest integral time. Ruelle argued many years ago that the growth of in-finitesimal perturbations in turbulence is ruled by the fastest timescale [1]. This leads to the prediction that the Lyapunov exponent is proportional to the inverse of the Kolmogorov time, and hence it increases with the Reynolds number. Turbulent flows at high Re are therefore strongly chaotic [2]. Nonetheless, the time that it takes for a small perturbation to affect significantly the dynamics of the large scales is expected to be of the order of the slow integral time [3]. The ratio between these extreme timescales increases with the Reynolds number and therefore allows a finite predictability time to coex-ists with strong chaos [4]. This is evident from everyday experience: while the Kolmogorov time of the atmosphere (in the planetary boundary layer) is a fraction of a second [5] the weather is predictable for days. The study of the predictability problem in turbulence dates back to the pioneering works of Lorenz [3] and of Leith and Kraichnan [6, 7]. The main idea of those studies is that a finite perturbation at a given scale in the inertial range of turbulence grows with the characteristic time at that scale. Therefore, while an infinitesimal perturbation is expected to grow exponentially fast, finite perturbations grow only algebraic in time, making the predictability of the flow much longer. These ideas were applied to the predictability of decaying turbulence [8], two-dimensional turbulence [9, 10] and three-dimensional turbulence at moderate Reynolds numbers [11]. In this letter we investigate, on the basis of high-resolution direct numerical simulations, chaos in homogeneous-isotropic turbulence by measuring the growth of the separation between two realizations starting from very close initial conditions. In the limit of in-finitesimal separation we compute the leading Lyapunov exponent of the flow (rate of exponential growth of the separation [12]) and we find that it increases with the Reynolds number, but surprisingly faster than what predicted on dimensional grounds [1] and what observed in low-dimensional models of turbulence [13]. For larger separation we observe the transition to an algebraic growth of the error, in agreement with the predictions of closure models [7]. Finally, we discuss the relation between chaoticity and the predictability time of turbulence (defined as the average time for the perturbation to reach a given threshold) in terms of the finite-size generalization of the Lyapunov exponents. We consider the dynamics of an incompressible velocity field u(x, t) given by the Navier-Stokes equations ∂ t u + u · ∇u = −∇P + ν∆u + f , (1) where P is the pressure field and ν is the kinematic viscosity of the fluid. The term f represents a mechanical forcing needed to sustain the flow. In the following we will present results in which the forcing is a determin-istic forcing with imposed energy input [14, 15]. The Navier-Stokes is solved numerically by a fully parallel pseudo-spectral code in a cubic box of size L at resolution N 3 with periodic boundary conditions in the three directions. The main parameters of the simulations are reported in Table I and further details are found in the Supplementary Material. In presence of forcing and dissipation, the turbulent flow reaches a statistically steady state in which the energy dissipation rate ε = ν(∂ α u β) 2 is equal to the input of energy provided by the forcing (brackets indicate average over the physical space). The turbulent state is characterized by a Kolmogorov energy spectrum E(k) = Cε 2/3 k −5/3. The kinetic energy E = E(k)dk = (1/2)|u| 2 fluctuates around a constant mean value, which defines the typical intensity of the large scale flow U = (2E/3) 1/2. The integral time is defined as T = E/ε and the integral scale is L = U T. We performed a series of simulations at increasing Reynolds number Re = U L/ν. In order to ensure that the viscous range is resolved with the same accuracy in