- The effect of different nonlinearities (Ampère force and Hall effect) on the saturation of a magnetic field generated by flows of conducting fluid is studied by means of numerical simulations. A three‐fluid (i.e., ions, electrons, and neutral particles) model is considered. The velocity field of the neutral particles is a prescribed, deterministic, incompressible three‐dimensional field in the form of the Arnol’d–Beltrami–Childress (ABC) flow. The dynamics of the charged components of fluid is determined by two‐fluid magnetohydrodynamics when ion–neutral particle collisions are taken into account. Four typical regimes of the nonlinear evolution of the magnetic field, corresponding to different types of nonlinearities (Ampère force or Hall effect) and different types of collisions (ion–ion collisions or ion–neutral particle collisions) are found. The transitions between these regimes, the structure of the saturated magnetic field, and the evolution of the magnetic field in these regimes are studied. Scaling estimates of the level of the saturated magnetic field and conditions obtained for the different regimes of the magnetic field evolution are in agreement with the results of the numerical simulations. © 1995 American Institute of Physics.