Asymptotic approximations for convection onset with Ekman pumping at low wavenumbers

Ekman pumping is a phenomenon induced by no-slip boundary conditions in rotating fluids. In the context of Rayleigh–Bénard convection, Ekman pumping causes a significant change in the linear stability of the system compared with when it is not present (that is, stress-free). Motivated by numerical solutions to the marginal stability problem of the incompressible Navier–Stokes equation (iNSE) system, we seek analytical asymptotic solutions which describe the departure of the no-slip solution from the stress-free one. The substitution of normal modes into a reduced asymptotic model yields a linear system for which we explore analytical solutions for various scalings of wavenumber. We find very good agreement between the analytical asymptotic solutions and the numerical solutions to the iNSE linear stability problem with no-slip boundary conditions.