Axisymmetric pulse train solutions in narrow-gap spherical Couette flow

TL;DR

This paper numerically investigates flow patterns in narrow-gap spherical Couette flow, revealing new traveling wave pulse train solutions and bifurcation behaviors as the aspect ratio decreases, extending understanding of flow dynamics in such systems.

Contribution

It introduces novel pulse train traveling wave solutions in narrow-gap spherical Couette flow and details bifurcation sequences at smaller aspect ratios, expanding prior knowledge.

Findings

Pulse train solutions travel toward the equator and annihilate in phase-slip events.

Bifurcation sequences depend on the aspect ratio, with direct transition to traveling waves at smaller gaps.

Solutions can be symmetric or asymmetric, periodic or quasi-periodic.

Abstract

We numerically compute the flow induced in a spherical shell by fixing the outer sphere and rotating the inner one. The aspect ratio $\epsilon=(r_o-r_i)/r_i$ is set at 0.04 and 0.02, and in each case the Reynolds number measuring the inner sphere's rotation rate is increased to $\sim10\%$ beyond the first bifurcation from the basic state flow. For $\epsilon =0.04$ the initial bifurcations are the same as in previous numerical work at $\epsilon=0.154$, and result in steady one- and two-vortex states. Further bifurcations yield travelling wave solutions similar to previous analytic results valid in the $\epsilon\to0$ limit. For $\epsilon=0.02$ the steady one-vortex state no longer exists, and the first bifurcation is directly to these travelling wave solutions, consisting of pulse trains of Taylor vortices travelling toward the equator from both hemispheres, and annihilating there in distinct phase-slip events. We explore these time-dependent solutions in detail, and find that they can be both equatorially symmetric and asymmetric, as well as periodic or quasi-periodic in time.