Bifurcation of polygonal spiral patterns in wide-gap spherical Couette flow

In spherical Couette flow between rotating inner and fixed outer concentric spheres, the radius ratio η = r_in / r_out determines the laminar-turbulent transition scenario. For narrow gaps (small η), the transition starts with Taylor vortices formed near the equatorial zone. On the other hand, for wide gaps (large η), the transition triggered by cross-flow instability around mid-latitude generates secondary wave with spiral vortices. We searched for the Reynolds numbers that spiral states exist in a wide gap case η = 0.5. We developed Newton method code for Navier-Stokes equation of spherical Couette flow and specified the spiral states. In the present study, we found that 3-fold spiral state and 4-fold spiral state coexist over the critical Reynolds number. It was confirmed that the 4-fold spiral state exists at a smaller Re than 3-fold spiral state. We also found that the periodic-like state is composed of the 3- and 4-fold spiral states, as if it is a beat of two different frequencies.