Instability by localized disturbances in critical region in a precessing sphere

The flow state in a precessing sphere of radius a is characterized by the spin angular velocity ?(s), the precession angular velocity ?(p) and the kinematic viscosity ? of fluid. The spin and precession axes are assumed to be orthogonal to each other. When two non-dimensional parameters, e.g. the Reynolds number Re & xfffd;=& xfffd;a(2)?(s)/? and the Poincar & xfffd; number Po & xfffd;=& xfffd;?(p)/?(s) are in some range of values, the flow is believed to approach a steady state even if it starts from an arbitrary initial condition. Such a stable region for the steady flow in the whole parameter space (Re, Po) is, however, not known yet. Here, we investigate, by the linear stability analysis of disturbances localized in a critical region, the boundary of the stable region of the steady flow in the strong spin and weak precession limit. It is found that the boundary curve takes the power form, Po=28.36Re?4/5Po?Re?1/2?1