A note on stabilising and destabilising effects of Ekman boundary layers

Marginal instability of a Bernard layer is considered in the asymptotic case of large rotation rate, i.e., for E --> 0, where E = nu/Ohm d(2) is the Ekman number, nu is the kinematic viscosity, Ohm is the angular velocity and d is the depth of the layer. The nature of the convection is determined by the magnitude of the Prandtl number Pr = nu/kappa, where kappa is the thermal diffusivity. The cases Pr > O(E) are studied here; the remaining possibility of thermoinertial waves arising when Pr less than or equal to O(E) has recently been analysed by us elsewhere (Phys. Fluids 9, 1980-1987, 1997). Attention is focused here on the role of the Ekman boundary layer in determining the critical value, R-c, of the Rayleigh number, R, at which convection is marginally possible, where R = g alpha beta d(2)/Ohm kappa; g is the gravitational acceleration, beta is the applied temperature gradient and alpha is the thermal expansion coefficient. For Pr greater than or equal to O(1), the marginal state is steady convection and R-c similar to E-1/3[3(2 pi(2))(2/3) - 4k(2 pi(2))(1/3) E-1/6] as E --> 0, where k = 0 when both the bounding surfaces are stress-free, k = 1 when one surface is stress-free and the other is non-slip, and k = 2 when both the bounding surfaces are nonslip. For O(E) < Pr < O(1), the marginal state is oscillatory convection and R-c similar to E-1/3[6(2 pi(2))Pr-2/3(4/3)/(1 + Pr)(1/3) + 2(4/3)k(3 + 5Pr) {root 2 pi Pr/(1 + Pr)}(2/3) E-1/6] as E --> 0. The second terms in these expressions for R-c, representing Ekman layer corrections to the leading order E-1/3 terms, can be substantial because of the small exponent of E. In steady convection, viscous dissipation in the Ekman layers destabilises the Bernard layer, but in oscillatory convection it is stabilising.