However, positive and negative charges
in the solar wind are differently
trapped in the van Allen belts, from
which they can then descend to the
Earth's atmosphere, which implies that
a net charge can be developed due to
this differential leakage. This begs
the question of whether there are any
estimates of the total net charge.
I've hunted in the literature but have
found little useful material other
than Dolezalek's 1988 paper:
Right I see an important misunderstanding here!
Positive and negative charges from the solar wind are NOT differently trapped. When we talk about the van Allen belts we are talking about high energy particles (MeV); low energy particles are still there, though in the majority of the outer belt the density is very low. Its fair to say that the electrons and protons go through different acceleration processes (its also worth noting that there are two inner belts, a positive and a negative that overlap as well as sitting on the region of the magnetosphere called the plasmasphere - a cold dense plasma that extends to about 4 Earth radii)
Some of the same waves that scatter relativistic waves into the loss cone and cause them to precipitate (electromagnetic ion-cyclotron waves) also scatter lower energy protons into the atmosphere. Over very short time scales there will be a charge difference, but on average quasi-neutrality will be maintained.
When precipitation occurs you have to remember that you are dealing with the ionosphere which is itself a plasma and therefore quasi-neutral. Current systems do form due to the collisions of ions with the neutral atmosphere. In the D-region, where MeV electrons deposit when they precipitate (in fact they cause extra ionisation and extend the D-layer downwards) there is a whole mixed bag of high collisions, attachment processes and recombination processes involving all sorts of complex ion-chemistry, the less energetic protons will deposit at higher altitudes so there would be a small charge gains in different regions but due to high conductivity in the E layer (120 km) and possible current systems that would not be the case for long. *
Now the atmosphere stuff is not my bag but you might like to go and look at the work of Dr. Martin Fullerkrug of the university of Bath who does some very cool work on sprites and whatnot.
Hope that helps
*I am not saying that exactly equal numbers of protons and electrons precipitate at a given moment but looking at the system over even relatively small time-scales the net charge would be damn close to zero.