The development of trapped structures in decaying and saturated drift wave turbulence is studied via computer simulation. A two-dimensional electrostatic fluid model is used. The turbulence that evolves in the pure decay runs (i.e., no nonadiabatic electrons) is characterized by tightly bound monopole vortices and a very narrow frequency spectrum. For the studies of saturated turbulence, a new nonadiabatic electron model is introduced, which gives the qualitatively correct response to a coherent trapped structure. This model takes into account the effects of broadening and shifting of the frequency spectrum. These effects are found to be quite important. Trapped structures are observed in many of the saturated simulations, even in the presence of moderately broad frequency spectra. The extent of the trapping varies dramatically, becoming a much stronger effect as the average electric field increases.