An asymptotic theory for wall modes in rapidly-rotating Rayleigh-Benard convection

At low Ekman number (Ek), the onset of convection is via wall modes: instabilities that are localized to the vertical boundaries. The wall modes lead to important heat transfer and fluid motions, even when the convection is sufficiently supercritical that both wall and bulk modes are present. While wall modes have been studied experimentally and via direct numerical simulations, neither approach can reach the low Ek of geophysical flows, e.g., convection in the Earth's outer core. Here we use multiple-scale asymptotic analysis to derive a set of integro-differential equations capturing the dynamics of wall modes in the Ek->0 limit. These equations match ``bulk'' variables, which satisfy geostrophic and thermal wind balance, to ``boundary layer'' variables which vary on a small length scale O(Ek^1/3) in the vertical side-wall thermal boundary layers. The boundary layer problem can be solved to leading order, yielding a non-linear and non-local boundary condition for the bulk variables. We solve these equations in cylindrical geometry using Dedalus for different reduced Rayleigh numbers R=Ra Ek, which asymptotically remains O(1). The numerical simulations match different qualitative and quantitative features observed in finite-Ek DNS and experiments, including front-like features which propagate around the cylinder at a characteristic velocity. Intriguingly, our simulations show this propagation velocity decreases with increasing R, even reversing direction for high supercriticality.