A comparison of 3D fluid, particle and hybrid model for negative streamers

Modeling individual free electrons can be important in the simulation of discharge streamers. Stochastic fluctuations in the electron density can accelerate the branching of streamers. In negative streamers, energetic electrons can even 'run away' and contribute to processes such as terrestrial gamma-ray and electron flashes. To track energies and locations of single electrons in relevant regions, we have developed a 3D hybrid model that couples a particle model for single electrons in the region of high fields and low electron densities with a fluid model in the rest of the domain. Here we validate our 3D hybrid model on a 3D (super-)particle model for negative streamers without photoionization in overvolted gaps. We show that the extended fluid model approximates the particle and the hybrid model well until stochastic fluctuations become important, while the classical fluid model underestimates velocities and ionization densities. We compare density fluctuations and the onset of branching between the models, and we compare the front velocities with an analytical approximation.