Frequency staircases in narrow-gap spherical Couette flow

This is the author accepted manuscript. The final version is available from Taylor & Francis via the DOI in this record.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.This paper was inspired by AMS’ attendance (19–21 March 2014) of the LMS Society Conference “Nonlinear stability theory: from weakly nonlinear theory to the verge of turbulence” to celebrate the 85th birthday of Professor J.T. Stuart. AMS gained further perspectives from attending (24–27 March 2014) the KITP program “Wave-flow interaction in geophysics, climate, astrophysics, and plasmas” at UCSB, where this research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. We are grateful to the referees for their helpful comments

Accurate approximations for planetary and gravity<strong> </strong>waves in a polar basin

The eigenfrequencies of freely propagating divergent barotropic planetary and gravity waves in a sphericalpolar cap are discussed. The key amplitude equation is derived with the full spherical geometry maintainedand leads to a second-order differential equation with coefficients functions of the co-latitude. Previous studyof this problem has derived approximations to the requisite frequencies by evaluating these coefficients atsome chosen fixed value of the co-latitude thereby reducing the problem to that of a constant coefficientdifferential equation solved easily using routine methods. Here, we demonstrate that such a simplification canbe avoided since the full equation can be solved by standard asymptotic methods based on the latitudinallimit of the polar basin as the natural small parameter. Three-term asymptotic series are developed which arein remarkably good accord with numerical solutions of the full equation

On the magnetic fields generated by experimental dynamos

We review the results obtained by three successful fluid dynamo experiments and discuss what has been learnt from them about the effect of turbulence on the dynamo threshold and saturation. We then discuss several questions that are still open and propose experiments that could be performed to answer some of them.Comment: 40 pages, 13 figure

The linear impulse response for disturbances in an oscillatory Stokes layer

© 2015 The Authors.A brief review is given of numerical simulation results for the evolution of disturbances in a flat oscillatory Stokes layer. A spatially localised form of impulsive forcing is applied to trigger the disturbances. For the linearized case, the disturbance development displays an intriguing family tree-like structure, which involves the birth of successive generations of wavepackets. Although some features of the wavepacket behaviour may be accounted for using a Floquet linear stability analysis, for Fourier modes with prescribed wavenumbers, the discovery of the tree-like structure was completely unexpected. Simulation results were also obtained for nonlinear disturbances, where highly localised spikes were found to develop