An inhomogeneous Landau equation with application to spherical Couette flow in the narrow gap limit

A nonlinear evolution equation is derived which governs 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. The tendency for vortices off the equator to oscillate introduces the phenomenon of phase mixing. Steady finite amplitude solutions of the model equation are determined both numerically and analytically. The linear eigensolutions are identified by an ascending sequence of eigenvalues λ=λn (n=0, 1, …). Each eigensolution has its own finite amplitude continuation under variation of λ which is a measure of the excess Taylor number. Without phase mixing each vortex amplitude increases monotonically with growing λ and does so indefinitely. In contrast, with phase mixing each pair of linear modes (n=0, 1), (n=2, 3), etc. are connected under the finite amplitude continuation; both the mode amplitude and λ remain bounded. Though the small phase mixing results agree with the non-phase mixed ones up to moderately large λ, this range is of limited extent. As phase mixing is increased the solution space shrinks and the amplitude of the remaining solutions is strongly suppressed.