Wall modes and the transition to bulk convection in rotating Rayleigh-Bénard convection

We investigate states of rapidly rotating Rayleigh-Bénard convection in a cylindrical cell over a range of Rayleigh number 3×10^5≤ Ra ≤ 5×10^9 and Ekman number 10^-6≤ Ek ≤ 10^-4 for Prandtl number Pr = 0.8 and aspect ratios 1/5 ≤Γ≤ 5 using direct numerical simulations. We characterize, for perfectly insulating sidewall boundary conditions, the first transition to convection via wall mode instability and the nonlinear growth and instability of the resulting wall mode states including a secondary transition to time dependence. We show how the radial structure of the vertical velocity u_z and the temperature T is captured well by the linear eigenfunctions of the wall mode instability where the radial width of u_z is δ_u_z∼ Ek^1/3 r/H whereas δ_T ∼ e^-k r (k is the wavenumber of an laterally infinite wall mode state). The disparity in spatial scales for Ek = 10^-6 means that the heat transport is dominated by the radial structure of u_z since T varies slowly over the radial scale δ_u_z. We further describe how the transition to a state of bulk convection is influenced by the presence of the wall mode states. We use temporal and spatial scales as measures of the local state of convection and the Nusselt number Nu as representative of global transport. Our results elucidate the evolution of the wall state of rotating convection and confirm that wall modes are strongly linked with the boundary zonal flow (BZF) being the robust remnant of nonlinear wall mode states. We also show how the heat transport (Nu) contributions of wall modes and bulk modes are related and discuss approaches to disentangling their relative contributions.