Weakly nonlinear analysis of the onset of convection in rotating spherical shells

A weakly nonlinear study is numerically conducted to determine the behaviour near the onset of convection in rotating spherical shells. The mathematical and numerical procedure is described in generality, with the results presented for an Earth-like radius ratio. Through the weakly nonlinear analysis a Stuart–Landau equation is obtained for the amplitude of the convective instability, valid in the vicinity of its onset. Using this amplitude equation we derive a reduced order model for the saturation of the instability via nonlinear effects and can completely describe the resultant limit cycle without the need to solve initial value problems. In particular the weakly nonlinear analysis is able to determine whether convection onsets as a supercritical or subcritical Hopf bifurcation through solving only linear 2D problems, specifically one eigenvalue and two linear boundary value problems. Using this, we efficiently determine that convection can onset subcritically in a spherical shell for a range of Prandtl numbers if the shell is heated internally, confirming previous predictions. Furthermore, by examining the weakly nonlinear coefficients we show that it is the strong zonal flow created through Reynolds and thermal stresses that determines whether convection is supercritical or subcritical.