Pulse-train solutions of a spatially heterogeneous amplitude equation arising in the subcritical instability of narrow gap spherical Couette flow
type: Item
instance of: scholarly article
title: Pulse-train solutions of a spatially heterogeneous amplitude equation arising in the subcritical instability of narrow gap spherical Couette flow
zbMATH Open document ID: 1183.76717
author: Edward W. Blockley
author: Andrew P. Bassom
author: Andrew D. Gilbert
author: Andrew M. Soward
published in: Physica D
publication date: 2007-05-23T00:00:00Z
review text: We investigate some complex solutions \(a(x,t)\) of the heterogeneous complex-Ginzburg-Landau equation \[ \partial a/\partial t=[\lambda(x)+ix-| a| ^ 2]a+\partial^ 2a/\partial x^ 2, \] in which the real driving coefficient \(\lambda(x)\) is either constant or the quadratic \(\lambda(0)-\Upsilon^ 2_ \varepsilon x^ 2\). 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 \(\varepsilon\)) 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 \(\Upsilon_\varepsilon\), which depends on sphere angular velocity ratio, generally tends to zero with \(\varepsilon\). Though we report new solutions for \(\Upsilon_\varepsilon\neq 0\), our main focus is the physically more interesting limit \(\Upsilon_\varepsilon=0\). When \(\lambda =\text{const}\), uniformly bounded solutions of our CGL equation on \(-\infty<x<\infty\) have some remarkable related features, which occur at all values of \(\lambda\). Firstly, the linearised equation has no non-trivial neutral modes \(\overline a(x)\exp(i\Omega t)\) with any real frequency \(\Omega\) including zero. Secondly, all evidence indicates that there are no steady solutions \(\overline a(x)\) of the nonlinear equation either. Nevertheless, \textit{A. P. Bassom} and \textit{A.M. Soward} [J. Fluid Mech. 499, 277--314 (2004; Zbl 1163.76364)] identified oscillatory finite amplitude solutions, \[ a(x,t)=\sum_ {n\in\mathbb Z}\overline a(x-x_ n)\exp\{i[(2n+1)\Omega t+\phi_ n]\}, \] expressed in terms of the single complex amplitude \(\overline a(x)\), which is localised as a pulse on the length scale \(L_{\text{PS}}=2\Omega\) about \(x=0\). Each pulse-amplitude \(\overline a(x-x_ n)\exp(i\phi_ n)\) is identical up to the phase \(\phi_n=(-1)^n\pi/4\), is …
Mathematics Subject Classification ID: 76E07
Mathematics Subject Classification ID: 76E30
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 37N10
zbMATH DE Number: 5156595
zbMATH Keywords: CGL-equation
zbMATH Keywords: pulse-train
zbMATH Keywords: Couette flow
zbMATH Keywords: Taylor vortices
describes a project that uses: AUTO
MaRDI profile type: Publication
full work available at URL: j.physd.2007.01.005
OpenAlex ID: W2124023738
Wikidata QID: Q57940426
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DOI: 10.1016/J.PHYSD.2007.01.005
description: scientific article; zbMATH DE number 5156595