Efficient kinetic particle simulations of space charge limited emission in magnetically insulated transmission lines using reduced physics models

Abstract

We explore the use of reduced physics models for efficient kinetic particle simulations of space charge limited (SCL) emission in inner magnetically insulated transmission lines (inner MITLs), with application to Sandia National Laboratories’ Z machine. We propose a drift kinetic (guiding center) model of electron motion in place of a fully kinetic model and electrostatic-magnetostatic fields in place of electromagnetic fields. The validity of these approximations is suggested by the operational parameters of the Z machine, namely, current pulse lengths of order 100 ns compared with Larmor periods typically smaller than 10^{-11} s, typical Larmor radii of a few (tens) of microns (magnetic fields of tens to hundreds of Tesla) compared with MITL dimensions of a few centimeters, and transient time of light waves along the inner MITL of order a fraction of a nanosecond. Guiding center orbits eliminate the fast electron gyromotion, which enables the use of tens to hundreds of times larger time steps in the numerical particle advance. Electrostatic-magnetostatic fields eliminate the Courant-Friedrichs-Lewy (CFL) numerical stability limit on the time step and allow the use of higher grid resolutions or, alternatively, larger time steps in the fields advance. Overall, potential computational cost savings of tens to hundreds of times exists. The applicability of the reduced physics models is examined on two problems. First, in the simulation of space charge limited emission of electrons from the cathode surface due to high electric fields in a radial inner MITL geometry with a short load. In particular, it is shown that a drift kinetic-based particle-in-cell (PIC) model with electrostatic-magnetostatic fields is able to accurately reproduce well-known physics of electron vortex formation, spatially and temporally. Second, deeper understanding is gained of the mechanism behind vortex formation in this MITL geometry by considering an exemplar problem of an electron block of charge. This simpler setup reveals that the main mechanism of vortex formation can be attributed to pure drift motion of the electrons, that is, the (fully kinetic) gyromotion of the electrons is inessential to the process. This exemplar problem also suggests a correlation of the spatial dimensions of vortices to the thickness of the electron layer, as observed in SCL simulations. It also confirms that the electromagnetic nature of the fields does not play an essential role. Finally, an improved hybrid fully kinetic and drift kinetic model for electron motion is proposed, as means of capturing finite Larmor radius (FLR) effects; the particular FLR physics that is missed by the drift kinetic model is the particle-wall interaction. By initializing SCL emitted electrons as fully kinetic and later transitioning them to drift kinetic, according to simple criteria, the accuracy of SCL simulations can be improved, while preserving the potential for computational efficiency.