The instability of a layer of fluid heated below and subject to Coriolis forces

Abstract

This paper is devoted to examining the stability of a horizontal layer of fluid heated below, subject to an effective gravity (g) acting (approximately) in the direction of the vertical and the Coriolis force resulting from a rotation of angular velocity Ω about a direction making an angle \vartheta with the vertical. It is shown that the effect of the Coriolis force is to inhibit the onset of convection, the extent of the inhibition depending on the value of the non-dimensional parameter T = 4d⁴Ω² cos² \vartheta /ν², where d denotes the depth of the layer and ν is the kinematic viscosity. Tables of the critical Rayleigh numbers (R_c) for the onset of convection are provided for the three cases (a) both bounding surfaces free, (b) both bounding surfaces rigid and (c) one bounding surface free and the other rigid. In all three cases R_c→ constant × T^⅔ as T→∞ ; the corresponding dependence of the critical temperature gradient (-β_c) for the onset of convection, on ν and d, is gα β_c = constant × \ kappa (Ω⁴ cos⁴ \vartheta /d⁴ν)^⅓ (kappa is the coefficient of thermometric conductivity and a is the coefficient of volume expansion). The question whether thermal instability can set in as oscillations of increasing amplitude (i.e. as 'overstability') is examined for case (a), and it is shown that if kappa /ν< 1\cdot 478 this possibility does not arise; but if kappa /ν > 1\cdot 478, over-stability is the first type of instability to arise for all T greater than a certain determinate value. It further appears that these latter possibilities should be considered in meteorological and astrophysical applications of the theory.