Frequency staircases in narrow-gap spherical Couette flow

Abstract

Recent studies of plane parallel flows have emphasised the importance of finite-amplitude self-sustaining processes for the existence of alternative non-trivial solutions. The idea behind these mechanisms is that the motion is composed of distinct structures that interact to self-sustain. These solutions are not unique and their totality form a skeleton about which the actual realised motion is attracted. Related features can be found in spherical Couette flow between two rotating spheres in the limit of narrow-gap width. At lowest order the onset of instability is manifested by Taylor vortices localised in the vicinity of the equator. By approximating the spheres by their tangent cylinders at the equator, a critical Taylor number based on the ensuing cylindrical Couette flow problem would appear to provide a lowest order approximation to the true critical Taylor number. At next order, the latitudinal modulation of their amplitude a satisfies the complex Ginzburg-Landau equation (CGLe) ∂a/∂t=(λ+ix)a+∂2a/∂x2−|a|2a, ∂a/∂t=(λ+ix)a+∂2a/∂x2-|a|2a, where −x-x is latitude scaled on the modulation length scale, t is time and λλ is proportional to the excess Taylor number. The amplitude a governed by our CGLe is linearly stable for all λλ but possesses non-decaying nonlinear solutions at finite λλ, directly analogous to plane Couette flow. Furthermore, whereas the important balance ∂a/∂t=ixa∂a/∂t=ixa suggests that the Taylor vortices ought to propagate as waves towards the equator with frequency proportional to latitude, the realised solutions are found to exist as pulses, each locked to a discrete frequency, of spatially modulated Taylor vortices. Collectively they form a pulse train. Thus the expected continuous spatial variation of the frequency is broken into steps (forming a staircase) on which motion is dominated by the local pulse. A wealth of solutions of our CGLe have been found and some may be stable. Nevertheless, when higher-order terms are reinstated, solutions are modulated on a yet longer length scale and must evolve. So, whereas there is an underlying pulse structure in the small but finite gap limit, motion is likely to be always weakly chaotic. Our CGLe and its solution provides a paradigm for many geophysical and astrophysical flows capturing in minimalistic form interaction of phase mixing ixaixa, diffusion ∂2a/∂x2∂2a/∂x2 and nonlinearity |a|2a|a|2a.