Quantifying spatiotemporal chaos in Rayleigh-Bénard convection.

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-Bénard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading-order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading-order Lyapunov exponent, and we quantify the details of their response to the dynamics of defects. The leading-order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary-dominated dynamics to bulk-dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics.

[1]  Einar M. Rønquist,et al.  An Operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow , 1990 .

[2]  Extensive chaos in the nikolaevskii model , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  Paul Fischer,et al.  Pattern formation and dynamics in Rayleigh-Benard convection : numerical simulations of experimentally realistic geometries. , 2002, nlin/0210066.

[4]  Steven H. Strogatz,et al.  Nonlinear Dynamics and Chaos , 2024 .

[5]  Egolf,et al.  Lyapunov spectral analysis of a nonequilibrium Ising-like transition. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[7]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[8]  David A. Egolf.  Dynamical Dimension of Defects in Spatiotemporal Chaos , 1998 .

[9]  Antonio Politi,et al.  Dynamic localization of Lyapunov vectors in spacetime chaos , 1998 .

[10]  F. Busse,et al.  Non-linear properties of thermal convection , 1978 .

[11]  A Garfinkel,et al.  Controlling cardiac chaos. , 1992, Science.

[12]  Henry S Greenside,et al.  Microextensive chaos of a spatially extended system. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  E. Ott.  Chaos in Dynamical Systems: Contents , 1993 .

[14]  Paul Fischer,et al.  An Overlapping Schwarz Method for Spectral Element Solution of the Incompressible Navier-Stokes Equations , 1997 .

[15]  Le Gal P,et al.  Turbulence in a cylindrical container of argon near threshold of convection. , 1985, Physical review letters.

[16]  Gollub,et al.  Statistical studies of chaotic wave patterns. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  D. Ruelle.  Large volume limit of the distribution of characteristic exponents in turbulence , 1982 .

[18]  J. Yorke,et al.  Dimension of chaotic attractors , 1982 .

[19]  G. Ahlers,et al.  Low-Temperature Studies of the Rayieigh-Bénard Instability and Turbulence , 1974 .

[20]  H.M. Tufo,et al.  Terascale Spectral Element Algorithms and Implementations , 1999, ACM/IEEE SC 1999 Conference (SC'99).

[21]  Morris,et al.  Spiral defect chaos in large aspect ratio Rayleigh-Bénard convection. , 1993, Physical review letters.

[22]  Ilarion V. Melnikov,et al.  Mechanisms of extensive spatiotemporal chaos in Rayleigh–Bénard convection , 2000, Nature.

[23]  Revealing the building blocks of spatiotemporal chaos: deviations from extensivity. , 2006, Physical review letters.

[24]  Manuel G. Velarde,et al.  Convective instability: A physicist's approach , 1977 .

[25]  Steven A. Orszag,et al.  Stability and Lyapunov stability of dynamical systems: A differential approach and a numerical method , 1987 .

[26]  M. Cross,et al.  Nonlinear dynamics: Chaos in space and time , 2000, Nature.

[27]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[28]  F. Busse.  On the Stability of Two-Dimensional Convection in a Layer Heated from Below , 1967 .