Have you seen the auroras? Could you define them with equations? Fluid mechanics equations might come close enough for animations....i donno..
The thing for me is, we do not understand the cascade that produces electricity's effect.
So many theories, with each giving insight into a bit of the puzzle. We still use outmoded nomenclature and the word pictures that they associate with.
Take "bow shock", which refers to a wave front produced by an object traveling through Earth's atmosphere at high speed. If it is in the plasma of space, I call it a SMEDLY, Severe Magnetic Effect Deviation Layer Yanker .
Now SMEDLYS look like bow shocks, but they are as different as look a likes, praying mantis and a twig...