Asymptotic approximations for convection onset with Ekman pumping at low wavenumbers

TL;DR

This work analyzes how Ekman pumping from no-slip boundaries modifies the linear stability of rapidly rotating Rayleigh-Bénard convection beyond the stress-free case. By deriving a reduced composite quasi-geostrophic (CQG) model with a pumping boundary condition and performing an asymptotic marginal-stability analysis across four low-wavenumber regimes, the authors obtain explicit leading-order expressions for the marginal Rayleigh number $\widetilde{Ra}$ as a function of $k_\perp$ and show excellent agreement with full iNSE numerics in the interior. The results reveal distinct scaling laws and transitions among diffusion and buoyancy balances, including a high sensitivity of the marginal curve at small $k_\perp$ and a broad instability range for large $\widetilde{Ra}$, highlighting the non-vanishing influence of Ekman pumping as $\varepsilon\to 0$. These insights improve understanding of boundary-layer effects in rotating convection and have potential implications for interpreting simulations of geophysical and astrophysical flows where Ekman pumping cannot be neglected.

Abstract

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 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.

Figures (5)

• Figure 1: Marginal stability of the system in the $\widetilde{Ra}$-$k_\perp$ plane for $\varepsilon = 10^{-4}$ and $\sigma = 1$. The numerically computed curve for the iNSE system with Ekman pumping is shown in solid blue. The blue dash-dot curve is the stress-free case, $\widetilde{Ra} = k_\perp^4+n^2\pi^2/k_\perp^2$. The asymptotic approximations for cases (i)-(iv) are in red, purple, yellow, and green respectively.
• Figure 2: Eigenfunction profiles for case (i), $k_\perp^2\sim \varepsilon^{1/2}$. We have selected as an example $\varepsilon = 10^{-3}$ and $k_\perp = \varepsilon^{1/4}$. Numerically computed eigenfunctions to the iNSE problem are in solid blue, and the analytical asymptotic approximations (\ref{['eq:case1efuns']}) are in dashed red.
• Figure 3: Eigenfunction profiles for case (ii), $\varepsilon^{1/2}\gtrsim k_\perp^2 \gtrsim \varepsilon^2$, with $\varepsilon = 10^{-4}$ and $k_\perp \approx .0203$. Numerically computed eigenfunctions to the iNSE problem are in solid blue, and the analytical asymptotic approximations (\ref{['eq:case2efuns']}) are in dashed red.
• Figure 4: Eigenfunction profiles for case (iii), $k_\perp^2 \sim \varepsilon^2$, with $\varepsilon = 10^{-3}$, $k_\perp = 9\cdot 10^{-3}$. Numerically computed eigenfunctions to the iNSE problem are in solid blue, and the analytical asymptotic approximations (\ref{['eq:case3efuns']}) are in dashed red.
• Figure 5: Eigenfunction profiles for case (iv), $k_\perp^2 \lesssim \varepsilon^2$, with $\varepsilon = 10^{-3}$, $k_\perp = 10^{-4}$. Numerically computed eigenfunctions to the iNSE problem are in solid blue, and the analytical asymptotic approximations (\ref{['eq:case4efuns']}) are in dashed red.