The incompressible and anelastic Navier-Stokes equations are considered in the limit of rapid rotation (small Ekman number). The analysis is limited to horizontal scales small enough so that both horizontal and vertical velocities are comparable, but the horizontal velocity components are still in geostrophic balance. Asymptotic analysis leads to a pair of nonlinear equations for the vertical velocity and vertical vorticity coupled by vertical stretching. Statistically stationary states are maintained against viscous dissipation by thermal forcing. Direct numerical simulation of the reduced equations reveals the presence of intense vortical structures spanning the layer depth, in excellent agreement with recent experiments of Sakai (1997) and simulations of the Boussinesq equations for rotating convection by Julien, Legg, McWilliams & Werne (1996). Exact but fully nonlinear steady and oscillatory solutions of the reduced partial differential equations are found for various two- and three-dimensional planforms.