An energetically consistent Rayleigh number to characterise heterogeneous thermal forcing based on available potential energy.

Studies of convection involving spatially heterogeneous heating and cooling typically adopt ad hoc definitions of the Rayleigh number by postulating length and temperature scales to characterise destabilising effects. Such definitions are problematic for making consistent comparisons and do not always have a well-defined physical justification. We therefore propose a precise definition of the Rayleigh number that has a consistent physical meaning across a range of problems involving spatially heterogeneous heating and cooling. The definition is based on a proxy for the supply of available potential energy, where the latter is positive semidefinite by construction and quantifies the provision of potential energy that can theoretically be converted into kinetic energy. The definition is therefore able to quantify the extent to which an arbitrary spatial distribution of heating or cooling is likely to be `destabilising'. We apply the definition to a wide range of examples and demonstrate that it extends naturally to a general formulation of such problems involving heating and cooling that is specified as a joint probability distribution in space and/or time. Along the way we clarify the connection between the concept of available potential energy and problems in optimal transport.