Double-diffusive convection with a non-linear equation of state: Part I. The accurate conservation of properties in a two-layer system

If the equation of state is nonlinear, a given flux of heat across a double diffusive interface causes different buoyancy fluxes in the upper and lower layers. This results in different convective activity in the two layers and can lead to preferential entrainment across the interface in one direction (i.e. a migration of the interface). In this paper we derive the conservation equations for properties (e.g. heat and a solute) across a double diffusive interface between two well-mixed layers. A nondimensional measure of the entrainment across an interface and the most suitable choice for the buoyancy flux ratio are presented. Some surprising facts emerge. First, even for a linear equation of state and in the absence of direct entrainment across the interface, the flux of water across a finger interface is shown to be important. Second, for the heat-solute system, the heat balance equations for each well-mixed layer contain terms proportional to the heat of solution of the solute and the partial specific enthalpy of pure water in a seawater solution. Third, the rate of change of gravitational potential energy of the two-layer system is shown to have several extra terms in addition to the two commonly quoted major terms.