In plasma bowls, the plasma filaments made of ionized inert gases extend from the central 'mini Tesla coil', all the way to the glass outer sphere. What makes them want to go to the glass? Why don't they simply form some other random shapes that don't touch the glass?
So this is actually a question which is not very well-studied! Princeton appears to have put out a paper about it in 2010 (PDF) with some concrete answers gleaned from watching these globes with high-speed cameras, so that's something nice; in 2013 the lead author also did a teleseminar with some slides(PDF).
So the basic physics is that the outside world is electrically neutral (zero voltage) and the inside bulb is being rapidly varied from a very high positive voltage to a very high negative voltage, and electrons like to move from negative voltages to positive ones. So as the central core goes negative, the electrons want to fly outward to the glass bulb; as the central core goes positive, the electrons want to fly back. But, the gas in the ball is not able to conduct these electrons intrinsically -- gases are normally electrical insulators. But pretty much every electrical insulator becomes a conductor if you put enough voltage across it -- what happens is that the atoms of the insulator eventually start getting torn apart by the desire for the electrons to go one way and the nuclei to go the other, and we enter this new state of matter, "plasma", where some electrons get shared among multiple nuclei, able to conduct electricity over them. This is called "dielectric breakdown." The dielectric breakdown goes from inside to outside basically because the voltage field is approximately spherically symmetric -- the dominant force on the electrons is either being pulled straight towards the center or being pushed straight away from it, so when the plasma ionizes the electrons are being pushed or pulled directly away from the center, and that's what causes the arcs to go to the outside.
One of the simple results gained by looking at these filaments with a high-speed camera is that actually these plasma filaments are only visible for about 15% of each voltage cycle and are not initially visible at all. The plasma starts off being a sort of diffuse "glow" around the sphere, however tiny irregularities in which parts are getting more or less ionized turn out to be slightly better or worse conductors when the plasma is switched on/off.
So what you're seeing is what we'd call in physics "hysteresis", an effect where the history of a system impacts what it does after. The gas inside the ball has these certain pathways which have been -- at first by random chance -- slightly better at conducting electrons towards the outside than others. When the plasma reforms, those gas molecules get preferentially ripped apart first, and this is why filaments persist over thousands of cycles and can be seen by the naked eye.
So the plasma globe is actually showing you "tracks where electrons have recently been moving" and these tracks look like solid entities but actually they're flickering faster than your eye can see. The electrons are moving outward and inward to the center based on these very fast back-and-forth forces on them, and that's why these tracks go from the center to the edge.