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

We investigate some complex solutions a(x, t) of the heterogeneous complex-Ginzburg-Landau equationpartial derivative a/partial derivative t = [lambda(x) + ix - vertical bar a vertical bar(2)]a + partial derivative(2)a/partial derivative x(2),in which the real driving coefficient lambda(x) is either constant or the quadratic lambda(0) - gamma(2)(epsilon)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 epsilon) 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 empty set(epsilon), which depends oil sphere angular velocity ratio, generally tends to zero with epsilon. Though we report new solutions for empty set(epsilon not equal) 0, our main focus is the physically more interesting limit empty set(epsilon) = 0.When lambda = constant, uniformly bounded solutions of our CGL equation on -infinity < x < infinity have some remarkable related features, which occur at all values of lambda. Firstly, the linearised equation has no non-trivial neutral modes a(x) exp(i ohm t) with any real frequency Q 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 How, J. Fluid. Mech. 499 (2004) 277-314. Referred to as BS] identified oscillatory finite amplitude solutions,[GRAPHICS]expressed in terms of the single complex amplitude a(x), which is locilised as a pulse on the length scale L-PS = 20 about x = 0. Each pulseamplitude a(x - x(n)) exp(i phi(n)) is identical up to the phase phi(n) = (-1)(n)pi/4, is centred at x(n) = (n + 1/2)L-PS and oscillates at frequency (2(n) + 1)ohm. 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, in excess of some threshold lambda(MIN) (> 0), solutions with pulse-separation L-PS were located on a finite range L-min(lambda) <= L-PS <= L-max(lambda).Here, we seek new pulse-train solutions, for which the product a (x, t) exp(-ixt) is spatially periodic on the length 2L = NLPS, N is an element of N. The BS-mode at small lambda has N = 2, and on increasing lambda 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 lambda, while that preference yields to the other symmetry-broken N = 2 solution at larger lambda. 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. (C) 2007 Elsevier B.V. All rights reserved.