We present numerical simulations of steady, laminar, axisymmetric convection of a Boussinesq fluid in a shallow, rotating, cylindrical domain. The flow is driven by an imposed vertical heat flux and shaped by the background rotation of the domain. The geometry is inspired by that of tropical cyclones and the global flow pattern consists of a shallow swirling vortex combined with a poloidal flow in the r -z plane which is predominantly inward near the bottom boundary and outward along the upper surface. Our numerical experiments confirm that, as suggested in our recent work [L. Oruba et al., J. Fluid Mech. 812, 890 (2017), 10.1017/jfm.2016.846], an eye forms at the center of the vortex which is reminiscent of that seen in a tropical cyclone and is characterized by a local reversal in the direction of the poloidal flow. We establish scaling laws for the flow and map out the conditions under which an eye will, or will not, form. We show that, to leading order, the velocity scales with V =(αg β ) 1 /2H , where g is gravity, α is the expansion coefficient, β is the background temperature gradient, and H is the depth of the domain. We also show that the two most important parameters controlling the flow are Re =V H /ν and Ro =V /(Ω H ) , where Ω is the background rotation rate and ν the viscosity. The Prandtl number and aspect ratio also play an important, if secondary, role. Finally, and most importantly, we establish the criteria required for eye formation. These consist of a lower bound on Re , upper and lower bounds on Ro , and an upper bound on the Ekman number.