On New Localized Vortex Solutions in the Couette-Ekman Layer

Abstract

Transition to turbulence can often be understood through analysis of instabilities and bifurcations. Plane Couette-flow and Hagen-Poiseuille flow, however, are well known to be linearly stable for all Reynolds numbers. Rather recently finite-amplitude states not bifurcating from the laminar base flow have been found [1, 2] and are now being discussed as possibly forming decisive structures in the high-dimensional phase space in which the transition to turbulence actually takes place [3, 4, 5]. It is noteworthy that most of the recently investigated finite-amplitude states in plane Couette and Hagen-Poiseuille flow are periodic and infinitely extended in the spatial directions in which the underlying system is homogeneous. One of the remaining questions seems to be, however, how the spatial localization of turbulence structures like puffs and patches can be explained in terms of a dynamical systems perspective. The existence of localized finite-amplitude states in the style of strongly nonlinear dissipative solitons is an attractive hypothesis in that context: The complex spatially localized turbulence could then be affected in one way or another by the phase-space properties and the phase space neighborhood of such localized solutions. The existence of these localized, solitary solutions in plane Couette flow is still being debated [6, 7, 8, 9]. In other pattern forming systems solitary solutions have, however, been found: in magnetohydrodynamics so-called ‘convectons’ have recently been detected [10, 11]; in Ekman type flows localized vortex solutions have been reported [12]; and also in granular materials spatially strongly localized patterns have been observed [13].