Pulse-train solutions of a spatially-heterogeneous amplitude equation arising in the subcritical instability of narrow-gap spherical Couette flow
Blockley, EW and Bassom, AP and Gilbert, AD and Soward, AM, Pulse-train solutions of a spatially-heterogeneous amplitude equation arising in the subcritical instability of narrow-gap spherical Couette flow, Physica D - Nonlinear Phenomena, 228, (1) pp. 1-30. ISSN 0167-2789 (2007) [Refereed Article]
in which the real driving coefficient λ(x) is either constant or the quadratic . This CGL equation arises in the weakly nonlinear theory of spherical Couette flow between two concentric spheres caused by rotating them both about a common axis with distinct angular velocities in the narrow gap (aspect ratio ε) limit. Roughly square Taylor vortices are confined near the equator, where their amplitude modulation a(x,t) varies with a suitably ‘stretched’ latitude x. The value of Υε, which depends on sphere angular velocity ratio, generally tends to zero with ε. Though we report new solutions for Υε≠0, our main focus is the physically more interesting limit Υε=0.
When , uniformly bounded solutions of our CGL equation on −∞ have some remarkable related features, which occur at all values of λ. Firstly, the linearised equation has no non-trivial neutral modes with any real frequency Ω including zero. Secondly, all evidence indicates that there are no steady solutions of the nonlinear equation either. Nevertheless, Bassom and Soward [A.P. Bassom, A.M. Soward, On finite amplitude subcritical instability in narrow-gap spherical Couette flow, J. Fluid. Mech. 499 (2004) 277–314. Referred to as BS] identified oscillatory finite amplitude solutions,
expressed in terms of the single complex amplitude , which is localised as a pulse on the length scale about x=0. Each pulse-amplitude is identical up to the phase ϕn=(−1)nπ/4, is centred at and oscillates at frequency (2n+1)Ω. The survival of the pulse-train depends upon the nonlinear mutual interaction of close neighbours; self-interaction is inadequate, as the absence of steady solutions shows. For given constant values of λ in excess of some threshold , solutions with pulse-separation were located on a finite range .
Here, we seek new pulse-train solutions, for which the product is spatially periodic on the length