Self organization in 2D circular shear layers

Experiments in forced circular shear layers performed both in magnetized plasmas and in rotating fluids reveal qualitatively similar bifurcation cascades involving states of circular vortex arrangements of varying complexity. We present results of numerical and asymptotic studies of the Navier-Stokes equations with Ekman forcing in an annular geometry that closely reproduce experimental observations. While stable to radially symmetric perturbations at any value of the Reynolds number Re, the steady state flow becomes unstable to azimuthal perturbations at a critical value Re_c. There ensues a braid-like arrangement of vortices straddling the forcing region and rotating at constant angular velocity. As Re is increased, these vortices grow like (Re - Re_c)^1/2 and eventually undergo a symmetry breaking transition to a new arrangement of fewer vortices. Further transitions can be observed as well as superpositions of various azimuthal modes with nontrivial temporal behavior. Linear stability analysis is performed to predict the first transition, and its results are found to be in close agreement both with direct simulations of the flow as well as with experimental observations. A weakly nonlinear analysis of the saturation of the instability is presented.