The non-linear behaviour of the collisionless drift instabilities and the resultant anomalous plasma diffusion have been studied by means of computer simulations and analytical theory. The simulation model used is a full three-dimensional electrostatic model of cylindrical geometry in an external magnetic field. Full dynamics is employed for ion motion while guiding-centre approximations are used for electron motion, which allows us to use rather realistic plasma parameters in the simulations. The simulation results indicate that a strong turbulence develops through non-linear interaction of the drift instabilities, which results in the formation of convective cells and anomalous particle diffusion. The broad frequency spectrum resembles that observed recently in toroidal confinement devices. Analytical theory is developed, based on the modecoupling process, to explain the generation of convective cells and strong plasma turbulence, along with an estimate of the resultant particle diffusion.