Three-dimensional oscillatory convection in a gravitationally modulated fluid layer

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Abstract
The equations for three-dimensional, time-dependent convection in a gravitationally modulated fluid layer heated from below are solved numerically using the Galerkin method in space and a Crank–Nicolson scheme in time. Nonlinear solutions are obtained for the Prandtl number of air (0.71) and for two Rayleigh numbers above the value for onset of oscillatory convection. Multiples of the fundamental frequency of oscillatory convection were chosen in order to study the effects of possible resonances of the frequency of gravitational modulation. Modulation causes a transition from traveling wave convection, which persists in the unmodulated case, to standing wave convection and phase locking occurs for moderate values of the amplitude of the dimensionless gravitational modulation (scaled with the standard acceleration of gravity) in the range 0 to 3. For larger values of the modulation amplitude, frequency locking breaks down and chaotic time dependence occurs.