An inhomogeneous Landau equation with application to spherical Couette flow in the narrow gap limit
Harris, D and Bassom, AP and Soward, AM, An inhomogeneous Landau equation with application to spherical Couette flow in the narrow gap limit, Physica D - Nonlinear Phenomena, 137, (3-4) pp. 260-276. ISSN 0167-2789 (2000) [Refereed Article]
In a previous paper [Physica D 137 (2000) 260], 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 narrowgap 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. The system is characterised by two parameters: λ, which is proportional to the Taylor number, and κ, which provides a measure of phase mixing. Only when the inner and outer spheres almost co-rotate is κ of order unity; otherwise κ is large. In [Physica D 137 (2000) 260], it was shown that there is a finite amplitude steady solution branch at fixed κ that connects the first two bifurcation points λ0 and λ1. For sufficiently large κ, the branch lies on λ0 ≤ λ ≤ λ1, but for smaller κ it extends beyond λ1; there on λ1 < λ < λN a large and small amplitude solution co-exist and coalesce at the nose of the branch, λN. In this paper we investigate both analytically and numerically the stability of the steady solutions and their subsequent evolution. Two types of modes exist - one (SP) preserves the reflectional symmetry of the steady solutions with respect to the equatorial plane, while the other (SB) breaks it. Three SP-globalbifurcation scenarios are identified. Each lead to limit cycles, which correspond to vortices drifting towards the equator from both sides. For small κ, a heteroclinic connection is made between the steady nose solution and its reversed flow state (opposite sign). The same occurs for moderate κ except that two oppositely signed small amplitude steady solutions are connected. For large κ a homoclinic cycle forms joining the undisturbed state to itself and this leads to a gluing bifurcation. This homoclinic cycle evolves from the vacillating wave limit cycle shed by an SP-Hopf bifurcation of the large amplitude solution. An SB-pitchfork bifurcation of the steady solutions leads to asymmetric drifting-phase solutions (travellingwaves). Time-stepping reveals that they are generally unstable and evolve into larger amplitude periodic solutions, which for small and moderate κ are SP-states. For large κ, the SB-drifting-phase solutions are strongly subcritical. The realised larger amplitude periodic state, to which they evolve, may be either of SB- or SP-type depending on the value of λ. These complicated solutions consist of trains of stationary pulses each modulating travellingwaves with distinct frequencies. The asymmetric SB-waves correspond to vortices drifting across the equator; yet far from it, where these vortices are very weak, they drift towards the equator as in the case of the SP-waves.