Convective instability in a two-layer fluid heated uniformly from above

Abstract

A fluid in which the dynamic and thermodynamic coefficients (μ, α, χ, cp etc.) vary vertically may become unstable when it is heated uniformly from above, assuming that the basic stratification is gravitationally stable and that the thermal expansion coefficient is positive. This “anti-convection” is studied in the special case of a fluid composed by two semi-infinite layers, each of which has constant coefficients. In the theory the Boussinesq approximation is used for each layer separately, and the interface is assumed to be plane. These assumptions seem justified in the present problem. Instability is favoured when the thermal expansion coefficient and the conductivity are large in one of the layers and small in the other layer. A necessary condition for instability is that the ratio μ1 α1Cp2/μ2 α2Cp1 is larger than 9 (or smaller than 1/9), where μ is the dynamic viscosity, α the thermal expansion coefficient, and cp the specific heat per unit mass. When this condition is fulfilled instability will occur for sufficiently large values of the ratio ?1/?2 (small values, respectively), where ? is the thermometric conductivity. The convection cells are confined to a region next to the interface. Small cells have comparable width and height, while larger cells are flattened in proportion to the 1/6 power of the Rayleigh number. Because of counteracting non-linear effects the convective motion cannot grow very large, the vertical velocities can at most be of order ?/D, where D is the scale of the cell. It is suggested that cells with Rayleigh numbers of order unity dominate the picture in finite amplitude situations. The driving energy comes from a region next to the boundary in the fluid layer having the larger thermal expansion and conductivity. Any motion started in the other layer creates thermal perturbations that are conducted into this region inducing a reversed circulation. When the thermal expansion coefficient is large this reversed circulation becomes strong. Through the frictional coupling it can directly amplify the primary motion and an instability develops. It should be possible to observe the instability in a laboratory experiment. A mercurywater system seems suitable for this purpose. The theoretical analysis shows that instability occurs when heating is applied at temperatures below 18°C. Geophysical and astrophysical applications seem possible. As example, the instability may appear at the interface between the atmosphere and the sea. An air-water system is stable under laboratory conditions, but in nature the existing eddy friction and conduction change the conditions considerably, and such as to favour the occurrence of the instability.DOI: 10.1111/j.2153-3490.1964.tb00171.x