The equations for a convective spherical shell are solved in Herring's approximation neglecting fluctuating self-interactions as an initial value problem. Free boundary conditions are used. The temperature field was expanded in spherical harmonics, YLm(θ,Φ), and the velocity field in poloidal vectors. The equations are m-independent, and YLm(θ,Φ) interacts only with YL−m(θ,Φ); the simplest convective modes are thus characterized by a value of L. The stability of the system was studied by perturbing an L-mode with (L ± 1)-modes and integrating the equations in time until it was reasonably certain that the perturbation was growing or decaying. For low Rayleigh numbers the stable mode is the same as the one that maximizes heat transport. Abstract The equations for a convective spherical shell are solved in Herring's approximation neglecting fluctuating self-interactions as an initial value problem. Free boundary conditions are used. The temperature field was expanded in spherical harmonics, YLm(θ,Φ), and the velocity field in poloidal vectors. The equations are m-independent, and YLm(θ,Φ) interacts only with YL−m(θ,Φ); the simplest convective modes are thus characterized by a value of L. The stability of the system was studied by perturbing an L-mode with (L ± 1)-modes and integrating the equations in time until it was reasonably certain that the perturbation was growing or decaying. For low Rayleigh numbers the stable mode is the same as the one that maximizes heat transport.