On Finite-amplitude Subcritical Instability in Narrow-gap Spherical Couette Flow

We consider the finite-amplitude instability of incompressible spherical Couette flow between two concentric spheres of radii R-1 and R-2 (>R-1) in the narrow-gap limit, epsilon = (R-2 - R-1)/R-1 much less than 1, caused by rotating them both about a common axis with distinct angular velocities Ohm(1) and Ohm(2) respectively. In this limit it is well-known that the onset of (global) linear instability is manifested by Taylor vortices of roughly square cross-section close to the equator. According to linear theory this occurs at a critical Taylor number Tit which, remarkably, exceeds the local value T, obtained by approximating the spheres as cylinders in the vicinity of the equator even as epsilon down arrow 0. Previous theoretical work on this problem has concentrated on the case of almost co-rotation with delta approximate to (Ohm(1) - Ohm(2))Ohm(1) = O(epsilon(1/2)) for which T-crit = T-c + O(delta(2)) + O(epsilon). In this limit the amplitude equation that governs the spatio-temporal modulation of the vortices on the latitudinal extent O(epsilon(1/2) R-1) gives rise to an interesting bifurcation sequence. In particular, the appearance of global bifurcations heralds the onset of complicated subcritical time-dependent finite-amplitude solutions.Here we switch attention to the case when epsilon(1/2) much less than delta less than or equal to 1. We show that for Taylor numbers T = T-c + O((deltaepsilon)(2/3)) there exists a locally unstable region of width O((deltaepsilon)R-1/3(1)) within which the amplitude equation admits solutions in the form of pulse-trains. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and consists of a wave propagating towards the equator under an envelope. The pulse drifts at a slow speed (relative to the wave velocity) proportional to its distance (and away) from the equator. Both the wavelength and the envelope width possess the same relatively short length scale O((epsilon(2)/delta)R-1/3(1)). The appropriate theory of spatially periodic pulse-trains is developed and numerical solutions found. Significantly, these solutions are strongly subcritical and have the property that T --> T-c as epsilon down arrow 0. Two particular limits of our theory are examined. In the first, epsilon(1/2) much less than delta much less than 1, the spheres almost co-rotate and the pulse drift velocity is negligible. A comparison is made of the pulse-train predictions with previously obtained numerical results pertaining to large (but finite) values of delta/epsilon(1/2). The agreement is excellent, despite the complicated long-time behaviour caused by inhomogeneity across the relatively wide unstable region.Our second special case delta = 1 relates to the situation when the outer sphere is at rest. Now the poleward drift of the pulses leads to a slow but exponential increase of their separation with time. This systematic pulse movement, over and above the spatial inhomogeneity just mentioned, necessarily leads to complicated and presumably chaotic spatio-temporal behaviour across the wide unstable region of width O(epsilon(1/3)R(1)) on its associated time scale, which is O(epsilon(-1/3)) longer than the wave period. In view of the several length and time scales involved only qualitative comparison with experimental results is feasible. Nevertheless, the pulse-train structure is robust and likely to provide the building block of the ensuing complex dynamics.