Lyapunov exponents for small aspect ratio Rayleigh-Bénard convection

Leading order Lyapunov exponents and their corresponding eigenvectors have been computed numerically for small aspect ratio, three-dimensional Rayleigh-Benard convection cells with no-slip boundary conditions. The parameters are the same as those used by Ahlers and Behringer [Phys. Rev. Lett. 40, 712 (1978)] and Gollub and Benson [J. Fluid Mech. 100, 449 (1980)] in their work on a periodic time dependence in Rayleigh-Benard convection cells. Our work confirms that the dynamics in these cells truly are chaotic as defined by a positive Lyapunov exponent. The time evolution of the leading order Lyapunov eigenvector in the chaotic regime will also be discussed. In addition we study the contributions to the leading order Lyapunov exponent for both time periodic and aperiodic states and find that while repeated dynamical events such as dislocation creation/annihilation and roll compression do contribute to the short time Lyapunov exponent dynamics, they do not contribute to the long time Lyapunov exponent. We find instead that nonrepeated events provide the most significant contribution to the long time leading order Lyapunov exponent

Scaling laws for rotating Rayleigh-Bénard convection

Numerical simulations of large aspect ratio, three-dimensional rotating Rayleigh-Bénard convection for no-slip boundary conditions have been performed in both cylinders and periodic boxes. We have focused near the threshold for the supercritical bifurcation from the conducting state to a convecting state exhibiting domain chaos. A detailed analysis of these simulations has been carried out and is compared with experimental results, as well as predictions from multiple scale perturbation theory. We find that the time scaling law agrees with the theoretical prediction, which is in contradiction to experimental results. We also have looked at the scaling of defect lengths and defect glide velocities. We find a separation of scales in defect diameters perpendicular and parallel to the rolls as expected, but the scaling laws for the two different lengths are in contradiction to theory. The defect velocity scaling law agrees with our theoretical prediction from multiple scale perturbation theory

Characterization of the domain chaos convection state by the largest Lyapunov exponent

Using numerical integrations of the Boussinesq equations in rotating cylindrical domains with realistic boundary conditions, we have computed the value of the largest Lyapunov exponent lambda1 for a variety of aspect ratios and driving strengths. We study in particular the domain chaos state, which bifurcates supercritically from the conducting fluid state and involves extended propagating fronts as well as point defects. We compare our results with those from Egolf et al., [Nature 404, 733 (2000)], who suggested that the value of lambda1 for the spiral defect chaos state of a convecting fluid was determined primarily by bursts of instability arising from short-lived, spatially localized dislocation nucleation events. We also show that the quantity lambda1 is not intensive for aspect ratios Gamma over the range 20<Gamma<40 and that the scaling exponent of lambda1 near onset is consistent with the value predicted by the amplitude equation formalism

Traveling waves in rotating Rayleigh-Bénard convection: Analysis of modes and mean flow

Numerical simulations of the Boussinesq equations with rotation for realistic no-slip boundary conditions and a finite annular domain are presented. These simulations reproduce traveling waves observed experimentally. Traveling waves are studied near threshhold by using the complex Ginzburg-Landau equation (CGLE): a mode analysis enables the CGLE coefficients to be determined. The CGLE coefficients are compared with previous experimental and theoretical results. Mean flows are also computed and found to be more significant as the Prandtl number decreases (from sigma=6.4 to sigma=1). In addition, the mean flow around the outer radius of the annulus appears to be correlated with the mean flow around the inner radius

Effect of the Centrifugal Force on Domain Chaos in Rayleigh-B\'enard convection

Experiments and simulations from a variety of sample sizes indicated that the centrifugal force significantly affects rotating Rayleigh-B\'enard convection-patterns. In a large-aspect-ratio sample, we observed a hybrid state consisting of domain chaos close to the sample center, surrounded by an annulus of nearly-stationary nearly-radial rolls populated by occasional defects reminiscent of undulation chaos. Although the Coriolis force is responsible for domain chaos, by comparing experiment and simulation we show that the centrifugal force is responsible for the radial rolls. Furthermore, simulations of the Boussinesq equations for smaller aspect ratios neglecting the centrifugal force yielded a domain precession-frequency $f\sim\epsilon^\mu$ with $\mu\simeq1$ as predicted by the amplitude-equation model for domain chaos, but contradicted by previous experiment. Additionally the simulations gave a domain size that was larger than in the experiment. When the centrifugal force was included in the simulation, $\mu$ and the domain size closely agreed with experiment.Comment: 8 pages, 11 figure

Casimir-Polder interaction between an atom and a small magnetodielectric sphere

On the basis of macroscopic quantum electrodynamics and point-scattering techniques, we derive a closed expression for the Casimir-Polder force between a ground-state atom and a small magnetodielectric sphere in an arbitrary environment. In order to allow for the presence of both bodies and media, local-field corrections are taken into account. Our results are compared with the known van der Waals force between two ground-state atoms. To continuously interpolate between the two extreme cases of a single atom and a macroscopic sphere, we also derive the force between an atom and a sphere of variable radius that is embedded in an Onsager local-field cavity. Numerical examples illustrate the theory.Comment: 9 pages, 4 figures, minor addition

Atomic multipole relaxation rates near surfaces

The spontaneous relaxation rates for an atom in free space and close to an absorbing surface are calculated to various orders of the electromagnetic multipole expansion. The spontaneous decay rates for dipole, quadrupole and octupole transitions are calculated in terms of their respective primitive electric multipole moments and the magnetic relaxation rate is calculated for the dipole and quadrupole transitions in terms of their respective primitive magnetic multipole moments. The theory of electromagnetic field quantization in magnetoelectric materials is used to derive general expressions for the decay rates in terms of the dyadic Green function. We focus on the decay rates in free space and near an infinite half space. For the decay of atoms near to an absorbing dielectric surface we find a hierarchy of scaling laws depending on the atom-surface distance z.Comment: Updated to journal version. 16 page