  The focus of this study is to investigate primary and secondary bifurcations
to weakly nonlinear flows (weak branch) in convective rotating spheres in a
regime where only strongly nonlinear oscillatory sub- and super-critical flows
(strong branch) were previously found in [E. J. Kaplan, N. Schaeffer, J. Vidal,
and P. Cardin, Phys. Rev. Lett. 119, 094501 (2017)]. The relevant regime
corresponds to low Prandtl and Ekman numbers, indicating a predominance of
Coriolis forces and thermal diffusion in the system. We provide the bifurcation
diagrams for rotating waves (RWs) computed by means of continuation methods and
the corresponding stability analysis of these periodic flows to detect
secondary bifurcations giving rise to quasiperiodic modulated rotating waves
(MRWs). Additional direct numerical simulations (DNS) are performed for the
analysis of these quasiperiodic flows for which Poincar\'e sections and kinetic
energy spectra are presented. The diffusion time scales are investigated as
well. Our study reveals very large initial transients (more than 30 diffusion
time units) for the nonlinear saturation of solutions on the weak branch,
either RWs or MRWs, when DNS are employed. In addition, we demonstrate that
MRWs have multimodal nature involving resonant triads. The modes can be located
in the bulk of the fluid or attached to the outer sphere and exhibit
multicellular structures. The different resonant modes forming the nonlinear
quasiperiodic flows can be predicted with the stability analysis of RWs, close
to the Hopf bifurcation point, by analyzing the leading unstable Floquet
eigenmode.
