The ambipolar diffusion approximation is used to model partially ionized plasma dynamics in a single-fluid setting. To correctly apply the commonly used version of ambipolar diffusion, a set of criteria should be satisfied including the requirement that the difference in velocity between charges and neutral species (known as drift velocity) is much smaller than the thermal velocity, otherwise the drift velocity will drive a non-negligible level of further collisions between the two species. In this paper, we explore the consequences of relaxing this assumption. We show that a new induction equation can be formulated without this assumption. This formulation reduces to the ambipolar induction equation in the case the drift velocity is small. In the large drift velocity limit, the feedback of the drift velocity on the collision frequency results in decreased diffusion of the magnetic field compared with the standard ambipolar diffusion approximation for the same parameters. This has a natural consequence of reducing the frictional heating that can occur. Applying this to results from flux emergence simulations where the expansion of the magnetic field leads to strong adiabatic cooling of the partially ionized chromosphere resulted in a noticeable reduction in the magnitude of the predicted drift velocities. This article is part of the theme issue 'Partially ionized plasma of the solar atmosphere: recent advances and future pathways'.
Keywords: ambipolar diffusion; magnetohydrodynamics; partially ionized plasma; prominence; solar chromosphere.