A three-dimensional spectral method based on a Chebyshev/Chebyshev/Fourier discretization is proposed to integrate the Navier-Stokes equations for natural convection flow with large temperature differences under the low Mach-number approximation. The generalized Stokes problem arising from the time discretization by a second-order semi-implicit scheme is solved by a preconditioned iterative Uzawa algorithm. The spectrally convergent algorithm is validated on well-documented Boussinesq and non-Boussinesq benchmarks. Finally, non-Boussineq convection is investigated in a tall differentially heated cavity of aspect ratio 8 with one homogeneous direction. The technique is shown to be efficient in terms of computing performances and accuracy. The study brings new stability results evidencing 3D effects compared to both Boussinesq convection and former two-dimensional non-Boussinesq solutions. In particular, for Ra ⩾ 105 two-dimensional solutions are shown to be unstable with respect to three-dimensional perturbations. A Görtler-type instability grows in the region of the curved streamlines at both ends of the cavity and gives rise to 3D steady and oscillatory solutions. At higher Rayleigh numbers, these vortices eventually interact with a boundary layer instability leading to complex dynamics with multiple steady and unsteady stable solutions.