Linear and nonlinear instabilities in rotating cylindrical Rayleigh-Bénard convection.

Linear and nonlinear convection in a rotating annular cylinder, under experimental boundary conditions, heated from below and rotating about a vertical axis are investigated. In addition to the usual physical parameters such as the Rayleigh and Taylor number, an important geometric parameter, the ratio of the inner to outer radius, enters into the problem. For intermediate ratios, linear stability analysis reveals that there exist two countertraveling convective waves which are nonlinearly significant: a retrograde wave located near the outer sidewall and a prograde wave adjacent to the inner sidewall. Several interesting phenomena of nonlinear convection are found: (i) tempospatially modulated countertraveling waves caused by an instability of the Eckhaus-Benjamin-Feir type, (ii) destructive countertraveling waves in which the existence or disappearance of the prograde wave is determined by its relative phase to the retrograde wave, and (iii) a saddle-node-type bifurcation in which the prograde wave takes an infinite amount of time to pass over the retrograde wave.

Hydrothermal convection in moderately thin spherical shells.

Hydrothermal convection of pore water with a temperature-dependent viscosity within a permeable, internally heated, moderately thin spherical shell is investigated by both a perturbation analysis and a direct numerical simulation. The analysis and simulation are mainly focused on a thin spherical shell in that convective instabilities are characterized by the spherical harmonic degree l=6 with a 13-fold mathematical degeneracy. Four different three-dimensional analytical solutions of convection are derived by removing the degeneracy through the nonlinear effect. A direct numerical simulation of the nonlinear problem is also carried out, showing satisfactory agreement between the analytical solutions and the numerical simulations.

Countertraveling waves in rotating Rayleigh-Bénard convection.

Linear and nonlinear counter-traveling waves in a fluid-filled annular cylinder with realistic no-slip boundary conditions uniformly heated from below and rotating about a vertical axis are investigated. When the gap of the annular cylinder is moderate, there exist two three-dimensional traveling waves driven by convective instabilities: a retrograde mode localized near the outer sidewall and a prograde mode adjacent to the inner sidewall with a different wave number, frequency and critical Rayleigh number. It is found that the retrogradely propagating mode is always more unstable and is marked by a larger azimuthal wave number. When the Rayleigh number is sufficiently large, both the counter-traveling modes can be excited and nonlinearly interacting, leading to an unusual nonlinear phenomenon in rotating Rayleigh-Bénard convection.

Nonlinear convection in rotating systems: slip-stick three-dimensional traveling waves.

We investigate convection in a fluid channel uniformly heated from below and rotating about a vertical axis. When the width of the channel is moderate, convective instabilities are characterized by two three-dimensional traveling waves having the same frequency and wave number but traveling in opposite directions with different spatial structures. This Rapid Communication demonstrates that neither the progradely nor the retrogradely traveling wave is physically realizable in the vicinity of the instability threshold. The nature of convection is marked by nonlinear interactions of the two oppositely traveling three-dimensional waves which interfere strongly, leading to either vacillating or stationary convective flows.

Multiplicity of nonlinear thermal convection in a spherical shell.

Linear and weakly nonlinear thermal convection in a moderately thin spherical shell in the presence of a spherically symmetric gravity subject to a spherically symmetric boundary condition is systematically investigated through fully three-dimensional numerical simulations. The convection problem is self-adjoint and the linear convective stability is characterized by l, the degree of a spherical harmonics Yml (theta,phi). While the radial structure of the linear convection is determined by the stability analysis, there exists a (2l + 1)-fold degeneracy in the horizontal structure of the spherical convection. When l = O(10) , i.e., in a moderately thin spherical shell, the removal or partial removal of the degeneracy represents a mathematically difficult, physically not well-understood problem. By starting with carefully chosen initial conditions, we are able to obtain a variety of nonlinear convective flows at exactly the same parameters near the onset of convection, including steady axisymmetric convection, steady azimuthally periodic convection, steady azimuthally nonperiodic convection, equatorially asymmetric convection, and steady convection in the form of a single giant spiral roll covering the whole spherical shell which is stable and robust for a wide range of the Prandtl number.

Patterns in spherical Rayleigh-Bénard convection: a giant spiral roll and its dislocations.

Thermal convection in a moderately thin spherical fluid layer in the presence of spherically symmetric gravity, spherical Rayleigh-Bénard convection, is investigated through fully three-dimensional numerical simulations. A steady spherical pattern in the form of a single giant spiral roll covering the whole spherical surface without defects is discovered near the onset of convection. Successive dislocations of the giant spiral roll are also found at larger Rayleigh numbers.