Leading-order analytical solutions of the marginal-stability problem for rapidly rotating Rayleigh-Benard convection with no-slip boundaries, covering the small-horizontal-wavenumber regime where Ekman pumping makes the onset curve depart from the classical stress-free result. Working from the composite quasi-geostrophic equations with the pumping boundary condition, four wavenumber scalings are treated in turn: k_perp^2 ~ eps^(1/2), where lateral viscous diffusion still balances vortex stretching and the eigenvalue satisfies a transcendental condition; eps^(1/2) >= k_perp^2 >= eps^2, where that balance is lost, the interior vertical velocity becomes constant, and the reduced Rayleigh number flattens to the constant 2*sqrt(2)/eps^(1/2); k_perp^2 ~ eps^2, where vertical and horizontal thermal diffusion balance and the thermal boundary layer becomes O(1); and k_perp^2 <= eps^2, where vertical diffusion dominates and the eigenvalue grows as 24*sqrt(2)*eps^(3/2)/k_perp^2. Each closed-form eigenvalue and eigenfunction is checked against numerical solutions of the incompressible Navier-Stokes equations, and the results yield an explicit estimate of the Rayleigh number above which no-slip and stress-free onset curves separate.
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 stabilit…