Route to hyperchaos in Rayleigh-Bénard convection

Abstract

-Transition to hyperchaotic regimes in Rayleigh-Bénard convection in a square periodicity cell is studied by three-dimensional numerical simulations. By fixing the Prandtl number at P = 0.3 and varying the Rayleigh number as a control parameter, a bifurcation diagram is constructed where a route to hyperchaos involving quasiperiodic regimes with two and three incommensurate frequencies, multistability, chaotic intermittent attractors and a sequence of boundary and interior crises is shown. The three largest Lyapunov exponents exhibit a linear scaling with the Rayleigh number and are positive in the final hyperchaotic attractor. Thus, a route to weak turbulence is found.

Magnetic field generation by intermittent convection

“…1(b) and Ra=2300 in fig. 1(c); 6 in [19]). For Ra ≥ 3000 the laminar phases in the intermittency are no longer recognizable at the time scale shown (see the kinetic energy evolution for attractors at Ra=3000, 4000 and 5000 in fig 1(d)).…”

“…We study magnetic field generation by the hyperchaotic convective attractors found in [19] for L = 4, P = 0.3 and 2160 ≤ Ra ≤ 5000 (the critical Rayleigh number for the onset of convection is Ra c =657.5 [23]). For a given value of Ra, the initial condition for the hydrodynamic part of the system, v(x, 0) and θ(x, 0), is taken from the corresponding convective attractor; the initial magnetic field is b(x, 0) = (cos(πx 2 /2), 0, 0) scaled such that the magnetic energy E b (0) = 10 −7 .…”

“…In recent multiscale analysis of the RBC dynamo problem in the presence of rotation [18], low Prandtl number convection was also found to be beneficial for magnetic field generation. In the present work, we follow our recently published analysis of transition to chaos and hyperchaos in RBC as a function of the Rayleigh number for P = 0.3 [19]. Chaotic systems are characterized by aperiodic motions that are sensitive to initial conditions, i.e., close initial conditions tend to locally diverge exponentially with time, with the mean divergence rate measured by a positive maximum Lyapunov exponent.…”

“…The intermittency in the velocity field described in ref. [19] as being due to global bifurcations constitutes the starting point for our search for an explanation for the intermittent dynamo. We stress that our hydrodynamic regimes are in a transition state between weak chaos and turbulence, therefore, the system displays a complex switching between highly different phases that is typically very difficult to be studied, since the convergence of average quantities is much slower than in fully developed turbulence, and the meaning of those averages may be rather deceptive if they do not take into account information about the different phases of the flow.…”

Lyapunov exponents of the Kuramoto--Sivashinsky PDE

“…The interplay between these contrasting features leads to significant spatio-temporal complexity (e.g., Hyman and Nicolaenko 1986;Cross and Hohenberg 1993;Cvitanović, Davidchack, and Siminos 2010), from intermittent disorder, through to chaos, hyperchaos and turbulence. Lyapunov exponents characterise this chaos and turbulence (e.g., Eckmann and Ruelle 1985;Ruelle and Takens 1971;Takens 1981), and are increasingly used to analyse such spatio-temporal complexity in various applications such as turbulent Poiseulle flow (Keefe, Moin, and Kim 1992), turbulence in flames (Hassanaly and Raman 2018), and Rayleigh-Bérnard fluid convection (Chertovskih, Chimanski, and Rempel 2015;Xu 2017).…”