There are other forces that can cause current to flow through the earth. Here is one that can last for hours or days:
In the 1970s scientists at the University of Alaska Fairbanks Geophysical Institute during aurora activity measured ten amperes of direct electrical current generated by the aurora in local power lines. Almost all electrical power lines are built to carry only alternating current. Direct current generated in these lines by the aurora causes heating and other problems in the equipment that is used to distribute alternating electrical power. As aurora occurred Geophysical Institute scientists also measured over 2000 amperes of direct current flowing in the Trans-Alaska Pipeline. This electrical current flows in and out of the pipeline where the pipeline touches the ground which causes increased corrosion or loss of metal from the pipeline at that point. Although the electrical current is large the voltage is only a few thousandths of a volt per kilometer, which is not enough to feel if you touch the pipe.
If I am ever in Alaska I am so not going swimming around that pipeline !!
While the current can flow in Earth, it is because of the solar wind, not the aurora, The aurora is the effect of the solar winds. in the peak of sun spot activity, the aurora can be seen very far south. we have seen it here in Indiana when a large solar flair happens and causes a Geo magnetic storm, Transformers explode and many strange things happen, but even then if you were to take voltage readings from two different points on Earth you would find that these events still didn't cause a difference of potential in earth, it causes a difference of potential between the earth and something influenced by the solar wind, such as the pipe line or the transmission wires. it was a big problem with the telegraph system, back in the early days, this was one of the reasons that an earth reference seemed so important.