Asymptotic approximations for convection onset with Ekman pumping at low wavenumbers

TL;DR

This paper develops analytical asymptotic solutions to describe how Ekman pumping affects the onset of convection in rotating fluids with no-slip boundaries, matching well with numerical results.

Contribution

It introduces novel asymptotic solutions for the linear stability problem of convection with Ekman pumping at low wavenumbers, extending understanding beyond stress-free conditions.

Findings

Analytical solutions agree with numerical stability results.

Ekman pumping significantly alters convection onset.

Asymptotic methods effectively model boundary effects.

Abstract

Ekman pumping is a phenomenon induced by no-slip boundary conditions in rotating fluids. In the context of Rayleigh-B\'enard convection, Ekman pumping causes a significant change in the linear stability of the system compared to when it is not present (that is, stress-free). Motivated by numerical solutions to the marginal stability problem of the incompressible Navier-Stokes (iNSE) system, we seek analytical asymptotic solutions which describe the departure of the no-slip solution from the stress-free. 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.