Techniques of theoretical analysis and numerical simulation are combined to produce a dynamical model of tropical cumulonimbus convection which features a close cooperation between the updraught and downdraught circulations. The cloud-scale dynamics determine the structure and transfer properties. Sub-cloud-scale transfer is unimportant. A steady-state dynamical model shows that the upshear or down-shear propagation speed, c, of a cumulonimbus cell relative to the mid-level flow is determined as a function of the convective available potential energy, CAPE, and weakly influenced by the windshear through a non-dimensional number, R, of the large-scale flow. This propagation speed is almost constant for a wide range of R, with c≃ 0.3CAPE, but only possible if R ≳ 2.8 (small shear). This contrasts with a previous result for another regime of convection, obtained by Moncrieff and Green (1972), if R ⩽ 1 (high shear). The transfer of momentum is distinctive and of large magnitude. The initiation and growth of the convective circulation represent an essentially nonlinear, finite-amplitude process, whose properties are closely related to the wind profile in the tropical atmosphere. The numerical simulations attain a quasi-steady state of a complex, three dimensional nature, basic features of which can be represented in terms of the steady-state analysis. Moreover, the outflow of the downdraught air is closely related to the attainment of a steady circulation and suggests a mechanism both for the maintenance and the eventual breakdown of the convective regime.