Thermal instabilities in the form of two-dimensional convection rolls in a rotating annulus with the flat rigid ends and a moderate gap are investigated by both asymptotic and numerical methods. It is shown that the thin Ekman boundary layers at the ends of the annulus, to which the convection rolls attach, play an active controlling role. An asymptotic theory for an asymptotically large Taylor numberTis developed to obtain complete analytical solutions of the convection rolls, indicating at leading orderRc=R0+C(T1/4/&lgr;),&ohgr;=0,where&lgr;is the aspect ratio,&ohgr;is the frequency,Rcis the critical Rayleigh number with the presence of the Ekman boundary layers andR0is the critical Rayleigh number without the influence of the boundary layers. WhileR0can be determined exactly by using Bessel functions as the eigenfunction, constantCis obtained by matching the interior convection rolls to explicit solutions of the Ekman boundary layers. In the corresponding numerical analysis, convection solutions in a rotating annulus are calculated up toT=108with&lgr;=1.The analytical and numerical convection solutions are then compared to show a remarkable quantitative agreement whenT>106.©1998 American Institute of Physics.