First results from a two-dimensional numerical study of the buoyant rise of twisted magnetic flux tubes in the solar convection zone are presented. We show in detail the process by which the transverse component of the field can suppress the splitting of the rising tube into two vortex filaments. For the suppression to be effective, the pitch angle of the twisted field lines has to be above a threshold given by the condition that the magnetic equivalent of the Weber number (see § 2.2) be below 1. The shape obtained for the tube and wake is strongly reminiscent of laboratory experiments with air bubbles rising in liquids. The magnetized region outside an equipartition boundary is peeled away from the tube: two sidelobes are formed, which lag behind the tube and contain only a fraction of the initial magnetic flux. This is similar to the formation of a skirt in the fluid dynamical case. The velocities of rise predicted by the thin flux tube approximation are compared with those obtained here.