Thermal instabilities of a contained fluid are investigated for a fairly general class of problems in which the dynamics are dominated by the effects of rotation. In systems of constant depth in the direction of the axis of rotation the instability develops when the buoyancy forces suffice to overcome the stabilizing effects of thermal conduction and of viscous dissipation in the Ekman boundary layers. Owing to the Taylor–Proudman theorem, a slight gradient in depth exerts a strongly stabilizing influence. The theory is applied to describe the instability of the ‘lower symmetric régime’ in the rotating annulus experiments at high rotation rates. An example of geophysical relevance is the instability of a self-gravitating, internally heated, rotating fluid sphere. The results of the perturbation theory for this problem agree reasonably well with the results of an extension of the analysis by Roberts (1968).

Thermal convection in a layer heated from below is an exemplary case for the study of non-linear fluid dynamics and the transition to turbulence. I n this review an outline is given of the present knowledge of the simplest realisation of convection in a layer of fluid satisfying the Oberbeck-Boussinesq approximation. Non-linear properties such as the dependence of the heat transport on Rayleigh and Prandtl numbers and the stability properties of convection rolls are emphasised in the discussion. Whenever possible, theoretical results are compared with experimental observations. A section on convection in rotating systems has been included, but the influence of other additional physical effects such as magnetic fields, side wall geometry, etc, has not been considered.

The stability of cellular convection flow in a layer heated from below is discussed for Rayleigh number R close to the critical value Rc. It is shown that in this region the stable stationary solution is determined by a minimum of the integral \[ \int_0^{H_0}R(H)\,dH, \] where R(H) is a functional of arbitrary convective velocity fields which satisfy the boundary conditions. For the stationary solutions R(H) is equal to the Rayleigh number. H0 is a given value of the convective heat transport. In a second part of the paper explicit results are derived for the convection problem with deviations from the Boussinesq approximation owing to the temperature dependence of the material properties.