This paper shows the influence of the inner radius on the stability and intensity of vertical vortices, qualitatively similar to dust devils and cyclones, generated in a cylindrical annulus non-homogeneously heated from below. Little relation is found between the intensity of the vortex and the magnitude of the inner radius. Strong stable vortices can be found for both small and large values of the inner radius. The Rankine combined vortex structure, that characterizes the tangential velocity in dust devils, is clearly observed when small values of the inner radius and large values of the ratio between the horizontal and vertical temperature differences are considered. A contraction on the radius of maximum azimuthal velocity is observed when the vortex is intensified by thermal mechanisms. This radius becomes then nearly stationary when frictional force balances the radial inflow generated by the pressure drop in the center, despite the vortex keeps intensifying. These results connect with the behavior of the radius of the maximum tangential wind associated with a hurricane.

The appearance, evolution, and disappearance of periodic and quasiperiodic dynamics of fluid flows in a cylindrical annulus locally heated from below are analyzed using nonlinear simulations. The results reveal a route of the transition from a steady axisymmetric vertical vortex to a chaotic flow. The chaotic flow regime is reached after a sequence of successive supercritical Hopf bifurcations to periodic, quasiperiodic, and chaotic flow regimes. A scenario similar to the Ruelle-Takens-Newhouse scenario is verified in this convective flow. In the transition to chaos we find the appearance of subvortices embedded in the primary axisymmetric vortex, flows where the subvortical structure strengthens and weakens, that almost disappears before reforming again, leading to a more disorganized flow to a final chaotic regime. Results are remarkable as they connect to observations describing formation, weakening, and virtual disappearance before revival of subvortices in some atmospheric swirls such as dust devils.

This work is an extension of the paper by Cea and Malanowski to the nonlocal and nonlinear framework. The addressed topic is the study of an optimal control problem driven by a nonlocal p-Laplacian equation that includes a coefficient playing the role of control in the optimization problem. The cost functional is the compliance, and the constraint on the states are of the Dirichlet homogeneous type. The goal of the present work is a numerical scheme for the nonlocal optimal control problem and its use to approximate solutions in the local setting. The main contributions of the paper are a maximum principle and a uniqueness result. These findings and the monotonicity properties of the p-Laplacian operator have been crucial to building an effective numerical scheme, which, at the same time, has provided the existence of optimal designs. Several numerical simulations complete the work.

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In this work, we study the development of vortical structures generated in a rotating cylinder non-homogeneously cooled on the top. In the axisymmetric regime, for moderate vertical temperature differences and any rotation rate, cyclonic and anticyclonic rotations coexist in the flow: a counterclockwise motion at upper levels, giving place to a vertical top-down vortex, and a clockwise rotation at lower levels that generates a spin up motion. For lower rotation rates and high enough vertical temperature differences, only cyclonic top-down vortices survive and get stronger. We perform a force balance analysis to explain the phenomena. In the non-axisymmetric regime, no anticyclonic rotation at the bottom is reported and the cyclonic top-down vortex either disappears or splits up in two top-down vortices, depending on the ambient rotation rate. The intensity of the cooling on the top and how localized this cool region is affect the flow developed. When the horizontal temperature difference on the top is larger than the vertical temperature difference between top and bottom, stable axisymmetric top-down vortices with an inner updraft of warmer air are reported. The more localized the cooling above, the more difficult the development of the inner updraft becomes. Results may contribute to the understanding of the relevance of thermal processes in tornadogenesis.