Centrifugal effects in rotating convection: Axisymmetric states and three-dimensional instabilities

Rotating convection is analysed numerically in a cylinder of aspect ratio one, for Prandtl number about 7. Traditionally, the problem has been studied within the Boussinesq approximation with density variation only incorporated in the gravitational buoyancy term and not in the centrifugal buoyancy term. In that limit, the governing equations admit a trivial conduction solution. However, the centrifugal buoyancy changes the problem in a fundamental manner, driving a large-scale circulation in which cool denser fluid is centrifuged radially outward and warm less-dense fluid is centrifuged radially inward, and so there is no trivial conduction state. For small Froude numbers, the transition to three-dimensional flow occurs for Rayleigh number R ≈ 7.5 × 10³. For Froude numbers larger than 0.4, the centrifugal buoyancy stabilizes the axisymmetric large-scale circulation flow in the parameter range explored (up to R = 3.5 × 10⁴). At intermediate Froude numbers, the transition to three-dimensional flow is via four different Hopf bifurcations, resulting in different coexisting branches of three-dimensional solutions. How the centrifugal and the gravitational buoyancies interact and compete, and the manner in which the flow becomes three-dimensional is different along each branch. The centrifugal buoyancy, even for relatively small Froude numbers, leads to quantitative and qualitative changes in the flow dynamics.