Tesla coil spark shape Why does the spark on a Tesla coil split evenly in a 3D Mercedes sign away from the feeding current wire in three even streamers? It is not easy to see at high voltages where the sparks are violent or where the spark has a path to an earth electrode but on a small Tesla coil with the spike sticking up the spark always splits evenly in three 120 degree streams and 120 degrees away from the feeding spike.
 A: Tesla Coil (TC) sparks can take on a wide variety of appearances depending on a variety of factors. In general, multiple air streamers repel each other due to electrostatic repulsion, since the outward-propagating streamer heads are driven from a common HV source. However, most spark splitting tends to consist of a single channel splitting into two branches rather than three (or more). 
Unfortunately, Tesla Coil (TC) streamers and leaders have not been rigorously studied in any formal fashion. In a TC, the combination of a toroid or spherical "topload" with significant isotropic capacity and a relatively large radius of curvature for high breakdown voltage, excited by a pulsed high-voltage RF source (the TC) having an increasing RF envelope, repetitive pulsed (typically at 100 - 400 "bangs" per second) all combine to create exceptionally long, open-air, unterminated (single-ended) discharges at comparatively low output terminal voltages. 
After initial breakdown at the HV terminal, TC open-air discharges are driven by displacement currents that have typical peak currents of amperes to tens of amperes for larger Tesla Coils. Displacement currents flowing between the topload and self-capacitance of the plasma discharges cause joule heating of the plasma channels. This allows higher-current main channels to remain hot, and partially conductive, between successive "bangs". The discharge from the current bang can build upon the existing hot channels from previous bangs, so a Tesla Coil discharge can grow in length over multiple bangs until overall spark losses match TC output power and dynamic equilibrium (and maximum spark length) is reached. Main discharge channels are bright blue-white and arc-like (leaders}, grading through lower-current branching sparks. The tips of the sparks grade into a more diffuse bluish glow made up of countless streamers and corona discharges that typically extend many 10's of cm beyond the tips of the discrete light-blue sparks. Higher-current roots and feeders are brighter and larger in diameter, becoming brightest and thickest in the main arc-like root that connects the entire discharge tree to the TC output terminal.  
As individual tips of discharges split, current in the parent channel is divided between the resulting branches. Splitting is thought to originate from instabilities at the tips of propagating streamers. The probability of any given tip splitting is higher for higher E-field gradients ahead of the streamer tips. So, by carefully controlling the shape of the RF envelope, it is possible to reduce or enhance tip splitting in TC sparks. Once a streamer splits, electrostatic repulsion between the heads of the separating streamer tips causes them to diverge from each other. A number of studies on 3D streamers reveal that the angle between the diverging branches has an approximate Gaussian distribution, centered at 43 degrees +/- about 12 degrees.  
An extreme version of spark propagation can occur when the amplitude of the HV RF envelope is carefully increased at a rate that permits spark growth with minimal splitting. Under these conditions a relatively low output voltage (only 50 - 60 kV peak at 200 - 350 kHz) creates sword-like discharges that can extend for 5 - 6 feet into the air. The overall spark path tends to loosely follow the E-field lines from the topload of a short, fat secondary coil and toroid. These systems are driven from a specially-designed pulsed, solid state power oscillator. In these systems, the pulse rate is much lower than most TC's, so every spark discharge is relatively independent from previous discharges:
Quasi-Continuous-Wave (QCW) pulsed solid-state TC
References:


*

*"Stereo-photography of streamers in air", S. Nijdam, J.S. Moerman, T.M.P. Briels, E.M. van Veldhuizen, U. Ebert, Appl. Phys. Lett. 92, 101502 (2008),  DOI: 10.1063/1.2894195

*"Multiple scales in streamer discharges, with an emphasis on moving boundary approximations", U Ebert, F Brau, G Derks, W Hundsdorfer, C-Y Kao, C Li, A Luque, B Meulenbroek, S Nijdam, V Ratushnaya, L Schafer and S Tanveer, IOP, Nonlinearity 24 (2011) C1–C26 doi:10.1088/0951-7715/24/1/C01
