Finite-amplitude convection rolls with integer wavenumbers

Abstract

Finite-amplitude convection in the form of convection rolls in a rapidly rotating annulus, driven by a radial gravity field, with stress-free flat ends and an arbitrary gap is investigated by perturbation analysis and by numerical simulations. In the small gap limit (neglecting the effect of curvature), the mathematical problem for aligned two-dimensional rolls becomes exactly the same as that of an infinitely extended fluid layer heated from below (the Rayleigh-Bénard problem). As a consequence of a moderate gap, however, nonlinear convection is characterized by a small number of discrete integer modes. The orientational degeneracy in the Rayleigh-Bénard problem is removed and the theoretical difficulties associated with the continuous unstable wavenumber band are also eliminated. Weakly nonlinear convection solutions are expressed in terms of the Bessel functions of the first and second kind. Numerical simulations focusing on an infinite Prandtl-number fluid are carefully checked with weakly nonlinear solutions. Two representative cases are examined in detail. The first case uses a rotating annulus with a small gap so that a connection with the Rayleigh-Bénard problem can be made. The second case uses a rotating annulus with a moderately large gap which shows fundamentally different nonlinear behaviours from those of the Rayleigh-Bénard problem.