Steady-State Model for Laser-Guided Lightning-Like Discharges

We have developed a reduced model for laser-guided discharges, which can be solved efficiently over the complete length and duration of the discharge, and also provides analytic insights. We assume the laser pre-pulse designates a long thin channel with given low conductivity, and that current flows entirely in this channel. Maxwell's equations can be reduced to a diffusion equation for E$_{z}$(z,t), with a diffusion coefficient D(z,t) proportional to the channel conductance, very small ahead of the discharge and rapidly increasing at the discharge head. By specifying that the discharge propagates at a constant speed u, the diffusion equation is further reduced to a first-order O.D.E. in $\tau \equiv $t--z/u, which must be solved self-consistently with rate equations that determine D($\tau )$. The required driving voltage pulse V($\tau )$ is an output of the calculation. Even in absence of deionization processes, we show the discharge propagates only if T$_{e}$ is large enough throughout the channel to drive continuing ionization. If the channel is narrow enough, this can occur at sustainable levels of E$_{z}$ due to saturation of the N$_{2}$ vibrational energy sink.