The Hasegawa - Wakatani (HW) model equations for resistive drift waves are investigated numerically and analytically. Two fundamentally different assumptions about the parallel electron dynamics have been investigated previously, and these are compared analytically and numerically. Analytically, it is found that the large-scale behaviour is different in the two cases. Numerically, it is found that the resulting quasi-two-dimensional turbulence is very sensitive to the assumption made, indicating that three-dimensional simulations are necessary for realistic simulation of the physical processes modelled by the HW system. In addition, a new numerical scheme for the time integration has enabled long-time simulations in the adiabatic regime showing that the total energy and the total generalized enstrophy do not vanish in this limit.