Air Breakdown (CHARGE)

The growth of the channel encompasses many regimes of plasma physics, from very weakly conducting air to fully ionized or doubly ionized regions. It also includes many different physical and atomic processes, involving may different species of particles, which are active at different times in the channel development. To model the development of an arc, we use fluid conservation equations [1]. The growth of the arc is well-described by the following equations [2] , especially in the early stages.

Particle density conservation

Momentum conservation

Where α is the index of the particle species, g is the ionization rate, ν_c is the collision frequency of the species with neutral particles, which is currently constant but can be a temperature dependent variable. For strongly ionized plasma this equation needs to include coulomb collisions and charge particle - neutral interactions.

Energy conservation

Where only electrons and heavy particles (all other air constituent than electrons) are considered, and the latter is assumed to have energy equally divided among species.

Additional consideration

The presence of molecular vibrational modes is accounted for in the term, which is the excess energy density in the electron fluid in a finite difference cell for one time step. The temperature dependent ionization rate is derived by Fowler [3] and we use his parameterized equations (and look-up tables) to avoid the numerical integration associated to the function derived by Fowler, see the figure below where Fowler’s ionization rate (y-axis) is provided as a function of electron temperature (x-axis) and electric field.

T. Rudolph and R. A. Perala, "Development and Application of Linear and Nonlinear Methods for Interpretation of Lightning Strikes to In-Flight Aircraft," NASA CR-3974, 1986.
T. Rudolph and R. Perala, "Linear and Nonlinear Interpretation of the Direst Strike Lightning Response of the NASA F-106B Thunderstorm Research Aircraft," NASA CR-3746, 1983.
R. Fowler, "A Trajectory Theory of Ionization in Strong Electric Fields," Journal of Physics B: At. Mol. Phys, 16, 1983.