Convective Space-Time Chaos as a Dynamical Model of Deterministic and Stochastic Turbulence

TL;DR

This work develops convective space-time chaos as a unifying dynamical model for deterministic and stochastic turbulence observed in open flows. It introduces two tractable models—the NHKS partial differential equation and a chain of nonlinear amplifiers (plus a coupled-map lattice)—and uses convective and chronotopic Lyapunov exponents to quantify growth and propagation of perturbations under inlet forcing. The study shows how random inlet forcing leads to noise-sustained turbulence alongside reliability via synchronization under common noise, and identifies the spatial Lyapunov exponent as a key measure of boundary-condition sensitivity, which governs the transition from deterministic to stochastic turbulence. It also analyzes periodic and quasiperiodic forcing, demonstrating how complexity grows downstream and can mimic randomness, and discusses space–time duality as a bridge between convective chaos and conventional chaos, with implications for plasma physics, electronics, and optics.

Abstract

Recently, a concept of deterministic and stochastic turbulence has been introduced based on experiments with a boundary layer. In these experiments, the flow was driven with controlled random perturbation; in addition, natural ambient noise was also present. Deterministic property manifested itself as repeatability of turbulence patterns induced by identical random perturbations at the inlet (deterministic turbulence). A stochastic non-identical component originating from natural noise grows and eventually dominates the flow further downstream (stochastic turbulence). We argue that these properties can be explained by exploring the concept of convective space-time chaos, where secondary perturbations on top of a chaotic state grow but move away in the laboratory reference frame. We illustrate this with two simple models of convective space-time chaos, one is a partial differential equation describing waves on a film flowing down a plate, and the other is a set of unidirectionally coupled ordinary differential equations. To prove convective space-time chaos, we calculate the profiles of the convective Lyapunov exponent. The repeatability of the turbulent field in different identical experimental runs corresponds to the reliability of stable dynamical systems in response to random forcing. The onset of the stochastic component is quantified with the spatial Lyapunov exponent. We demonstrate how an effective randomization of the field is observed when the driving is quasiperiodic. Furthermore, we discuss space-time duality, which links sensitivity to boundary conditions in the convective space-time chaos to the usual sensitivity to initial conditions in a standard chaotic regime.

Figures (13)

• Figure 1: Panel (a): a sketch of the experiments on controlled transition in a boundary layer (adapted after Figs. 1,2 in kachanov1994physical). Close to the inlet region, a ribbon band is embedded into the plate, allowing for the insertion of controlled perturbations in the flow. Panel (b): a sketch of the experiments on controlled transition for waves on a thin film flow over an inclined plate (adopted from Fig. 1 in liu1993measurements). A perturbation to the flow at the inlet is performed by applying pressure variations to the entrance manifold.
• Figure 2: Space-time diagrams of $u(x,t)$ for different $V$ and periodic boundary conditions, for $V=0$ (Eq. \ref{['eq:ks2']}) and $V=3$ (Eq. \ref{['eq:ks1']}).
• Figure 3: Convective LE for model \ref{['eq:ca']}. Blue line is fit $\lambda(v)=-1+1.305 v -0.905 v\ln v$. The dashed black line shows the construction of the spatial LE according to Eq. \ref{['eq:forsle']}.
• Figure 4: Panel (a): The chronotopic LE $\mathcal{L}(\mu;0)$. Markers: numerical results; solid line: polynomial fit $\mathcal{L}(\mu;0)=a-bx^2-cx^4-dx^6$ with $a = 0.0935; b=6.29; c=-39.4; d=219.7$. Panel (b): The convective Lyapunov exponent $\lambda(v;0)$. Markers: direct numerical calculations according to the definition \ref{['eq:convle']}, solid blue line - Legendre transform of the polynomial fit of $\mathcal{L}(\mu;0)$. The maximal speed of growing perturbations is $v_0\approx 1.4$.
• Figure 5: The autocorrelation functions for different spatial positions for an OU-process driving force in the NHKS model \ref{['eq:ks1']} with $V=2$. Panel (a): $x=0,2,4,6,8$, Panel (b): $x=10,12,14,16,18$, Panel (c): $x=20,\ldots,58$; Panel (d): $x=60,\ldots,78$.