The Küppers-Lortz instability occurs in rotating Rayleigh-Bénard convection and is a paradigmatic example of spatiotemporal chaos. Since the steady state of convection rolls is unstable to disturbance rolls oriented with an angle of about 60 degrees with respect to the given rolls in the prograde direction [G. Küppers and D. Lortz, J. Fluid Mech. 35, 609 (1969)], a spatiotemporally chaotic pattern is realized with patches of rolls continuously replaced by other patches in which the roll axis is switched by about 60 degrees. Surprisingly and contrary to this established scenario, Bajaj [Phys. Rev. Lett. 81 (1998)] observed experimentally square patterns in a cylindrical layer in the range of parameters where Küppers-Lortz instability was expected. In this paper we present square patterns which we have obtained in a numerical study by taking into account realistic boundary conditions. The Navier-Stokes and heat transport equations have been solved in the Oberbeck-Boussinesq approximation. The numerical method is pseudospectral and second order accurate in time. The rotation velocity of the square pattern increases linearly with the control parameter epsilon=Ra/R a(c) -1 , as in the experiment of Bajaj Furthermore, it was observed that this velocity decreases when the aspect ratio of the cylinder increases. These results indicate that the square pattern appears when the flow is laterally confined. The range of epsilon for which this pattern is stable tends to vanish for more extended layers.
The interplay between convective, rotational and magnetic forces defines the dynamics within the electrically conducting regions of planets and stars. Yet their triadic effects are separated from one another in most studies, arguably due to the richness of each subset. In a single laboratory experiment, we apply a fixed heat flux, two different magnetic field strengths and one rotation rate, allowing us to chart a continuous path through Rayleigh–Bénard convection (RBC), two regimes of magnetoconvection, rotating convection and two regimes of rotating magnetoconvection, before finishing back at RBC. Dynamically rapid transitions are determined to exist between jump rope vortex states, thermoelectrically driven magnetoprecessional modes, mixed wall- and oscillatory-mode rotating convection and a novel magnetostrophic wall mode. Thus, our laboratory ‘pub crawl’ provides a coherent intercomparison of the broadly varying responses arising as a function of the magnetorotational forces imposed on a liquid-metal convection system.
A horizontal fluid layer heated from below and rotating about a vertical axis in the presence of a vertical magnetic field is considered. From earlier work it is known that the onset of convection in a rotating layer usually occurs in the form of travelling waves attached to the vertical sidewalls of the layer. It is found that this behaviour persists when a vertical magnetic field is applied. When the Elsasser number Λ is kept constant and the sidewall is thermally insulating the critical Rayleigh number Rc increases in proportion to the rotation rate described by the square root of the Taylor number, τ. This asymptotic relationship is found for an electrically highly conducting sidewall as well as for an electrically insulating one. At fixed rotation rate for Q≫τ, Rc grows in proportion to Q when the sidewall is electrically highly conducting, and in proportion to Q3/4 when the sidewall is electrically insulating. Here Q is the Chandrasekhar number which is a measure of the magnetic energy density, and a thermally insulating sidewall has been assumed. Of particular interest is the possibility that the magnetic field counteracts the stabilizing influence of rotation on the onset of sidewall convection in the case of thermally insulating sidewalls. When the sidewall is thermally highly conducting, Rc for the sidewall mode grows in proportion to τ4/3. This asymptotic behaviour is found for both cases of electrical boundary conditions, but it no longer precedes the onset of bulk convection for Λ ≳ 1.
This paper deals with the optimal streaky perturbations ͑which maximize the perturbed energy growth͒ in a wedge flow boundary layer, which involves a favorable streamwise pressure gradient. These three-dimensional perturbations are governed by a system of linearized boundary layer equations around the Falkner-Skan base flow. Based on an asymptotic analysis of this system near the free-stream and the leading edge singularity, we show that for acute wedge semi-angle, all solutions converge after a streamwise transient to a single streamwise-growing solution of the linearized equations, whose initial condition near the leading edge is given by an eigenvalue problem first formulated in this context by Tumin ͓Phys. Fluids 13, 1521 ͑2001͔͒. Such a solution may be regarded as a streamwise evolving most unstable streaky mode, in analogy with the usual eigenmodes in strictly parallel flows, and shows an approximate self-similarity, which was partially known and is completed in this paper. An important consequence of this result is that the optimization procedure based on the adjoint equations heretofore used to define optimal streaks is not necessary. Instead, a simple low-dimensional optimization process is proposed and used to obtain optimal streaks. Comparison with previous results by Tumin and Ashpis ͓AIAA J. 41, 2297 ͑2003͔͒ shows an excellent agreement. The unstable streaky mode exhibits transient growth if the wedge semi-angle is smaller than a critical value that is slightly larger than / 6, and decays otherwise. Thus, the cases of right and obtuse wedge semi-angles exhibit less practical interest, but they show a qualitatively different behavior, which is briefly described in the supplementary material.
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