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

Abstract

We investigate some complex solutions a ( x , t ) of the heterogeneous complex-Ginzburg–Landau equation ∂ a / ∂ t = [ λ ( x ) + i x − | a | 2 ] a + ∂ 2 a / ∂ x 2 , in which the real driving coefficient λ ( x ) is either constant or the quadratic λ ( 0 ) − Υ ε 2 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 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 Υ ε , which depends on sphere angular velocity ratio, generally tends to zero with ε . Though we report new solutions for Υ ε ≠ 0 , our main focus is the physically more interesting limit Υ ε = 0 . When λ = constant , uniformly bounded solutions of our CGL equation on − ∞ < x < ∞ have some remarkable related features, which occur at all values of λ . Firstly, the linearised equation has no non-trivial neutral modes a ¯ ( x ) exp ( i Ω t ) with any real frequency Ω including zero. Secondly, all evidence indicates that there are no steady solutions a ¯ ( x ) 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, a ( x , t ) = ∑ n ∈ Z a ¯ ( x − x n ) exp { i [ ( 2 n + 1 ) Ω t + ϕ n ] } , expressed in terms of the single complex amplitude a ¯ ( x ) , which is localised as a pulse on the length scale L P S = 2 Ω about x = 0 . Each pulse-amplitude a ¯ ( x − x n ) exp ( i ϕ n ) is identical up to the phase ϕ n = ( − 1 ) n π / 4 , is centred at x n = ( n + 1 2 ) L P S and oscillates at frequency ( 2 n + 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 λ MIN ( > 0 ) , solutions with pulse-separation L P S were located on a finite range L min ( λ ) ≤ L P S ≤ L max ( λ ) . Here, we seek new pulse-train solutions, for which the product a ( x , t ) exp ( − i x t ) is spatially periodic on the length 2 L = N L P S , N ∈ N . The BS-mode at small λ has N = 2 , and on increasing λ it bifurcates to another symmetry-broken N = 2 solution. Other bifurcations to N = 6 were located. Solution branches with N odd, namely 3, 5, 7, were only found after solving initial value problems. Many of the large amplitude solutions are stable. Generally, the BS-mode is preferred at moderate λ , while that preference yields to the other symmetry-broken N = 2 solution at larger λ . Quasi-periodic solutions are also common. We conclude that finite amplitude solutions, not necessarily of BS-form, are robust in the sense that they persist and do not evaporate.