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Pulse-train solutions of a spatially-heterogeneous amplitude equation arising in the subcritical instability of narrow-gap spherical Couette flow

Citation

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]

Copyright Statement

Copyright 2007 Elsevier B.V.

DOI: doi:10.1016/j.physd.2007.01.005

Abstract

We investigate some complex solutions a(x,t) of the heterogeneous complex-Ginzburg–Landau equation

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in which the real driving coefficient λ(x) is either constant or the quadratic View the MathML source. 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 View the MathML source, 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 View the MathML source with any real frequency Ω including zero. Secondly, all evidence indicates that there are no steady solutions View the MathML source 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,

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expressed in terms of the single complex amplitude View the MathML source, which is localised as a pulse on the length scale View the MathML source about x=0. Each pulse-amplitude View the MathML source is identical up to the phase ϕn=(−1)nπ/4, is centred at View the MathML source 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 View the MathML source, solutions with pulse-separation View the MathML source were located on a finite range View the MathML source.

Here, we seek new pulse-train solutions, for which the product View the MathML source is spatially periodic on the length View the MathML source

Item Type:Refereed Article
Keywords:CGL-equation; Pulse-train; Couette flow; Taylor vortices
Research Division:Mathematical Sciences
Research Group:Applied Mathematics
Research Field:Theoretical and Applied Mechanics
Objective Division:Expanding Knowledge
Objective Group:Expanding Knowledge
Objective Field:Expanding Knowledge in the Mathematical Sciences
Author:Bassom, AP (Professor Andrew Bassom)
ID Code:107346
Year Published:2007
Web of Science® Times Cited:3
Deposited By:Mathematics and Physics
Deposited On:2016-03-11
Last Modified:2016-07-06
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