Convection in the presence of restraints

Abstract

In the presence of rotation or a magnetic field, the linearized convection problem reduces to a cubic characteristic equation. In part I, general methods are given for determining the onset of convection; in particular, the transition from oscillatory to steady modes is considered. The importance of this transition arises from evidence that oscillatory modes are inefficient at transporting heat. These methods are then applied to a rotating system where the critical Rayleigh number can be expressed in terms of a Taylor number. It is found that overstable modes develop into steady unstable modes before the exchange of stabilities for Prandtl numbers less than one-third. The nature of the motions is discussed and a similar treatment is provided for convection in a magnetic field. In part II, criteria for the onset of instability are derived from physical arguments. Convection can be treated by balancing the work done by buoyancy forces against the energy dissipated. In a rotating system, the effect of the Coriolis forces is to restrict the cell width and thus to enhance dissipation and promote stability. A magnetic field similarly attenuates the cells and prevents steady convection until the liberated kinetic energy exceeds the energy in the field. In part III, a cellular model is proposed for turbulent convection in a fluid of negligible viscosity, where the motion is limited by the non-linear transfer of energy to smaller-scale motions. If the Rayleigh number R the convective transport varies as R, while it varies as R. The discussion is extended to convection in the presence of rotation or a magnetic field; it is shown that overstable perturbations cannot develop into steady turbulent convection unless the system is already unstable to non-oscillatory modes. The transition from overstable to steady modes should therefore correspond to a sharp increase in convective transport.