Bifurcations of rotating waves in rotating spherical shell convection.

The dynamics and bifurcations of convective waves in rotating and buoyancy-driven spherical Rayleigh-Bénard convection are investigated numerically. The solution branches that arise as rotating waves (RWs) are traced by means of path-following methods, by varying the Rayleigh number as a control parameter for different rotation rates. The dependence of the azimuthal drift frequency of the RWs on the Ekman and Rayleigh numbers is determined and discussed. The influence of the rotation rate on the generation and stability of secondary branches is demonstrated. Multistability is typical in the parameter range considered.

Multistability in rotating spherical shell convection.

The multiplicity of stable convection patterns in a rotating spherical fluid shell heated from the inner boundary and driven by a central gravity field is presented. These solution branches that arise as rotating waves (RWs) are traced for varying Rayleigh number while their symmetry, stability, and bifurcations are studied. At increased Rayleigh numbers all the RWs undergo transitions to modulated rotating waves (MRWs) which are classified by their spatiotemporal symmetry. The generation of a third frequency for some of the MRWs is accompanied by a further loss of symmetry. Eventually a variety of MRWs, three-frequency solutions, and chaotic saddles and attractors control the dynamics for higher Rayleigh numbers.

Convection patterns in a spherical fluid shell.

Symmetry-breaking bifurcations have been studied for convection in a nonrotating spherical shell whose outer radius is twice the inner radius, under the influence of an externally applied central force field with a radial dependence proportional to 1/r(5). This work is motivated by the GeoFlow experiment, which is performed under microgravity condition at the International Space Station where this particular central force can be generated. In order to predict the observable patterns, simulations together with path-following techniques and stability computations have been applied. Branches of axisymmetric, octahedral, and seven-cell solutions have been traced. The bifurcations producing them have been identified and their stability ranges determined. At higher Rayleigh numbers, time-periodic states with a complex spatiotemporal symmetry are found, which we call breathing patterns.

Building a reduced model for nonlinear dynamics in Rayleigh-Bénard convection with counter-rotating disks.

A reduced model to decrease the number of degrees of freedom of the discretized Navier-Stokes equations to a small set that nevertheless captures the essential dynamics of the flow is proposed. The Rayleigh-Bénard convection problem in a cylinder of aspect ratio one where the lower and upper disks, maintained at hot and cold temperatures, respectively, rotate at equal and opposite angular velocities has been chosen to test the technique. The nonlinear dynamics is rich and complex when the temperature difference between disks and their angular velocity is varied. Representatives states--stationary, periodic near sinusoidal, and near heteroclinic--are presented. In each case, the reduced model is compared with temporal integration, and we show that 41 degrees of freedom are sufficient to reproduce the signal. We discuss the strengths and weaknesses of the algorithm by which we build our reduced model.

Scanning electrochemical microscopy: theory and application of the transient (chronoamperometric) SECM response.

A study of the transient (chronoamperometric) response of the scanning electrochemical microscope (SECM) is presented. SECM transients were simulated digitally with a novel integrator based on a Krylov algorithm. The transients observed with planar electrodes (PE), microdisks (MD), and thin-layer cells (TLC) are shown to be limiting cases that fit the simulated SECM transients at very short, intermediate, and long times, respectively. A procedure is established that, provided the tip radius is known, allows the determination of the diffusion coefficient of the species in solution independent of its concentration and the number of electrons transferred in the electrode reaction. Experimental SECM transients are reported for the electrochemical oxidation of Fe(CN)6(4-) in KCl; the diffusion coefficient of Fe(CN)6(4-) was found to agree very well with the literature value.