Fig. 1. Evolution of Urw1 on the (r, t) plane. It is obvious to find Urw1 exhibiting oscillations along r when t is very small.

Fig. 2. Profiles of Urw1 along r for different values of t. (a) Two extreme points are (0.851,0.233) and (1.507,0.419) for t=0.05; (b) there is no extreme point for t=0.25.

Fig. 3. Profiles of Urw1. (a) The isosurface of Urw1 at t=0.05 when Urw1=0.4; (b) the inside of (a) plotted from a bird’s-eye view and z∈[−0.9,0.9]; (c) the contour line of Urw1 at z=0. The latter two panels verify that there are three homocentric spheres.

Fig. 4. Radius of the sphere for the isosurface given by Urw1=0.5. Panel (b) is plotted for very small t of panel (a), and there is a minimum rmin=0.68574. Panel (c) is plotted for large t of panel (a), and there is a maximum rmax=2.00604. The asymptotical value of r is ras=2.

Fig. 5. Localized profiles of the Urw1 with z=0.5 in the (x, y) plane: (a) t=0; (b) t=0.5.

Fig. 6. Profiles of Urw1 with respect to t for different values of r: (a) r=0.25; (b) there are two extreme points (0.007,1.635) and (0.663,2.113) for r=0.5; (c) r=1.

Fig. 7. (a) Growth rates λ as a function of the spherical harmonic modes l. (b) Dominant unstable l=4 radial perturbation eigenmode emerging from a small random initial condition.