Phenomenological modelling of the first bifurcations of spherical Couette flow

The bifurcation of spherical Couette flow towards the Taylor-like one-vortex flow is analysed within the framework of a phenomenological model built using qualitative arguments based on symmetry and genericity considerations. Phase portraits of the corresponding two-dimensional dynamical system are presented and compared to results of numerical simulations of the axisymmetric Navier-Stokes equations.La bifurcation de l'écoulement de Couette sphérique vers l'écoulement à un rouleau de Taylor est analysée dans le cadre d'un modèle phénoménologique construit grâce à des arguments qualitatifs reposant sur des considérations de symétrie et de généricité. Les portraits de phase du système dynamique bidimensionnel correspondant sont présentés et comparés aux résultats de simulations numériques des équations de Navier-Stokes axisymétriques

Magnetohydrodynamics in a finite cylinder: Poloidal-toroidal decomposition

The magnetohydrodynamic equations present two challenging algorithmic requirements: that both fields be solenoidal and that the magnetic field match an unknown external field. The poloidal-toroidal decomposition represents a three-dimensional solenoidal vector field via two scalar potentials. Widely used in Cartesian and spherical geometries with periodic boundary conditions, complications appear in finite geometries which can, however, be circumvented. An implementation of the poloidal-toroidal decomposition for the magnetohydrodynamic equations in a finite cylinder is described, which uses a spectral spatial discretisation. A Green's function method is proposed for matching the magnetic field in a spectral representation to an external field in a vacuum

Self-sustaining process in Taylor-Couette flow

The transition from Tayor vortex flow to wavy-vortex flow is revisited. The self-sustaining process (SSP) of Waleffe [Phys. Fluids 9, 883 (1997)] proposes that a key ingredient in transition to turbulence in wall-bounded shear flows is a three-step process involving rolls advecting streamwise velocity, leading to streaks which become unstable to a wavy perturbation whose nonlinear interaction with itself feeds the rolls. We investigate this process in Taylor-Couette flow. The instability of Taylor-vortex flow to wavy-vortex flow, a process which is the inspiration for the second phase of the SSP, is shown to be caused by the streaks, with the rolls playing a negligible role, as predicted by Jones [J. Fluid Mech. 157, 135 (1985)] and demonstrated by Martinand et al. [Phys. Fluids 26, 094102 (2014)]. In the third phase of the SSP, the nonlinear interaction of the waves with themselves reinforces the rolls. We show this both quantitatively and qualitatively, identifying physical regions in which this reinforcement is strongest, and also demonstrate that this nonlinear interaction depletes the streaks

Complex dynamics in delay-differential equations with large delay

We investigate the dynamical properties of delay differential equations with large delay. Starting from a mathematical discussion of the singular limit τ → ∞, we present a novel theoretical approach to the stability properties of stationary solutions in such systems. We introduce the notion of strong and weak instabilities and describe a method that allows us to calculate asymptotic approximations of the corresponding parts of the spectrum. The theoretical results are illustrated by several examples, including the control of unstable steady states of focus type by time delayed feedback control and the stability of external cavity modes in the Lang-Kobayashi system for semiconductor lasers with optical feedback