A NOTE ON STABILIZING AND DESTABILIZING EFFECTS OF EKMAN BOUNDARY-LAYERS

Marginal instability of a Bernard layer is considered in the asymptoti c 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 angul ar velocity and d is the depth of the layer. The nature of the convect ion is determined by the magnitude of the Prandtl number Pr = nu/kappa , where kappa is the thermal diffusivity. The cases Pr > O(E) are stud ied here; the remaining possibility of thermoinertial waves arising wh en Pr less than or equal to O(E) has recently been analysed by us else where (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 po ssible, where R = g alpha beta d(2)/Ohm kappa; g is the gravitational acceleration, beta is the applied temperature gradient and alpha is th e 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 bot h the bounding surfaces are stress-free, k = 1 when one surface is str ess-free and the other is non-slip, and k = 2 when both the bounding s urfaces are nonslip. For O(E) < Pr < O(1), the marginal state is oscil latory 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 Ekma n layer corrections to the leading order E-1/3 terms, can be substanti al because of the small exponent of E. In steady convection, viscous d issipation in the Ekman layers destabilises the Bernard layer, but in oscillatory convection it is stabilising.