Plasma Instability Modeling With Mixed Wave/Diffusion Field Behavior

Abstract

Recent efforts have extended fluid dynamic modeling to include low-electrical-conductivity (wavelike) electromagnetic field effects in plasmas. Here, we present results of a new study to explore and illustrate the capability of resolving such effects in the development of plasma instabilities. Recent work[1, 2] has explored the development of an efficient numerical procedure for simulating the coupled fluid dynamic and full electromagnetic (Maxwell) equations. This system of equations is more general than the resistive magnetohydrodynamic (MHD) model. The physical advantage to this system is that the inclusion of the displacement current and charge separation effects extends fluid dynamic modeling of plasmas to include regions of extremely low (or zero) electrical conductivity, where the electromagnetic fields possess a wave nature. Capturing the correct physical behavior of plasma instabilities in the limit of extremely low electrical conductivity is of significance for modern aerospace applications, including energy bypass arrangements for scramjet propulsion using magnetohydrodynamic augmentation, due to the extremely low electrical conductivity of the aerospace environment. The results depicted in this paper illustrate the potential for the numerical method realized in Ref. 1 to effectively resolve this behavior in cases where the MHD approximation may not be valid due to an exceedingly low electrical conductivity. A new study was initiated to explore more fully the capabilities to resolve regions of both low and high electrical conductivity using this numerical scheme. The problem selected for this study was the classic MHD KelvinHelmholtz instability. The initial and boundary conditions are provided in Ref. 2. A periodic box is initialized with two fluids of different density moving in opposite directions. An initial small perturbation is applied to the interfaces between the fluids, seeding an instability. A 256x256 grid size using a Roe solver for the coupled system of Navier-Stokes and full Maxwell equations was used, as detailed in [1, 2]. Three cases were explored: Case A was assigned a zero electrical conductivity, σ = 0 (purely hydrodynamic), Case B was given a small electrical conductivity of σ = 1, and Case C was given a very high conductivity of σ = 10 6 . Case A provides a hydrodynamic benchmark. Case B permits an investigation of a simulation that includes the effects of electromagnetic wave propagation in the plasma evolution. Case C effectively realizes the magnetohydrodynamic approximation. Results for the simulations are presented in a unique fashion in Fig. 1. For each subfigure, three spacetime isosurfaces of the density (blue, green and orange corresponding to the color range) have been depicted. Isosurfaces are the same for all three presented subfigures. Time flows from right (initial conditions) to left (final solution). For Case A, the fluid achieves a highly chaotic solution. The propagation of the initial disturbance at the interfaces between the two fluids is clear, as well as their development into a highly turbulent flow. Case B possesses an extremely low electrical conductivity, and therefore is not subject to the magnetic diffusion approximation of MHD. In fact, electromagnetic wave propagation was visible in this solution. The presence of a finite electrical conductivity provides some diffusion, but it does not match very well to the MHD prediction. However, the conductivity is large enough to abate the low-wavelength instabilities somewhat, and provide stability to the flow. Case C realizes the magnetohydrodynamic limit, where the magnetic field tends to stabilize the flow and strongly suppresses low-wavelength instabilities. A much more organized series of flow structures evolve through time to the final state. The magnetic field is effectively “frozen” to the fluid with such a large electrical conductivity, and the field strongly diffuses with little wave propagation.