Patterns, Nonlinear Dynamics and Applications - PANDA

Andrew Soward

Global bifurcation to travelling waves with application to narrow gap spherical Couette flow

Abstract:

In a previous paper (Harris et al, 2000, Physica D 137, 260-276, [1]), an inhomogeneous complex Landau equation was derived in the context of the amplitude modulation of Taylor vortices between two rapidly rotating concentric spheres, which bound a narrow gap and almost co-rotate about a common axis of symmetry. In this weakly nonlinear regime the latitudinal vortex width is comparable to the gap between the shells. The vortices are located close to the equator and are modulated on a latitudinal length scale large compared to the gap width but small compared to the shell radius.

Here we investigate both analytically and numerically the stability and subsequent evolution of the steady finite amplitude solutions found in [1]. A complicated bifurcation structure is unravelled dependent on the magnitude k of spatial phase mixing. Only when the inner and outer spheres almost  corotate is k small; otherwise k is large. Two types of mode exist-- one (SP) preserves the reflectional symmetry of the steady solutions with respect to the equatorial plane while the other type (SB) breaks it. For sufficiently large k, a supercritical SP-Hopf bifurcation of the steady state leads to a vacillating solution which expands into a homoclinic cycle connecting the trivial undisturbed state to itself. SP-global bifurcations occur for all k leading to limit cycles which correspond to vortices drifting towards the equator from both sides. For small k a pair of heteroclinic connections is made between two steady states of equal amplitude but of opposite sign. As k is increased the steady state amplitude falls to zero leading for large k to the gluing of the aforementioned homoclinic cycles. For moderate k the nonlinear development shows no evidence of any stable symmetry-broken temporally periodic states although for large phase mixing k >> 1 strongly subcritical finite amplitude periodic SB-solutions are identified. On increasing the driving parameter (related to the Taylor number) at fixed large k both SP- and SB-periodic travelling wave solutions are obtained numerically. The asymmetric SB-waves correspond to vortices drifting across the equator although far from it, where these vortices are very weak, they drift towards the equator as in the case of the SP-waves.
 

This is joint work with

Derek Harris¹  (D.Harris@ex.ac.uk),

and Andrew P. Bassom^1,2 (A.P.Bassom@ex.ac.uk).

¹  School of Mathematical Sciences, University of Exeter, Exeter, EX4 4QE, UK.
²  School of Mathematics, University of New South Wales, Sydney 2052, Australia.