  The ambipolar diffusion approximation is used to model partially ionised
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 consequence 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 ionised chromosphere
resulted in a noticeable reduction in the magnitude of the predicted drift
velocities.
