Phase space eigenfunctions with applications to continuum kinetic simulations

Continuum kinetic simulations are increasingly capable of resolving high-dimensional phase space with advances in computing. These capabilities can be more fully explored by using linear kinetic theory to initialize the self-consistent field and phase space perturbations of kinetic instabilities. The phase space perturbation of a kinetic eigenfunction in unmagnetized plasma has a simple analytic form, and in magnetized plasma may be well approximated by truncation of a cyclotron-harmonic expansion. We catalogue the most common use cases with a historical discussion of kinetic eigenfunctions and by conducting nonlinear Vlasov-Poisson and Vlasov-Maxwell simulations of single- and multi-mode two-stream, loss-cone, and Weibel instabilities in unmagnetized and magnetized plasmas with one- and two-dimensional geometries. Applications to quasilinear kinetic theory are discussed and applied to the bump-on-tail instability. In order to compute eigenvalues we present novel representations of the dielectric function for ring distributions in magnetized plasmas with power series, hypergeometric, and trigonometric integral forms. Eigenfunction phase space fluctuations are visualized for prototypical cases such as the Bernstein modes to build intuition. In addition, phase portraits are presented for the magnetic well associated with nonlinear saturation of the Weibel instability, distinguishing current-density-generating trapping structures from charge-density-generating ones.Comment: 51 pages, 26 figures, 4 appendice

Whole Device Modeling of the FuZE Sheared-Flow-Stabilized Z Pinch

The FuZE sheared-flow-stabilized Z pinch at Zap Energy is simulated using whole-device modeling employing an axisymmetric resistive magnetohydrodynamic formulation implemented within the discontinuous Galerkin WARPXM framework. Simulations show formation of Z pinches with densities of approximately 10^22 m^-3 and total DD fusion neutron rate of 10^7 per {\mu}s for approximately 2 {\mu}s. Simulation-derived synthetic diagnostics show peak currents and voltages within 10% and total yield within approximately 30% of experiment for similar plasma mass. The simulations provide insight into the plasma dynamics in the experiment and enable a predictive capability for exploring design changes on devices built at Zap Energy.Comment: 8 pages, 9 figures, IAEA FEC 202

Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit

This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB) model of plasma physics. This model consists of the pressureless gas dynamics equations coupled with the Poisson equation and where the Boltzmann relation relates the potential to the electron density. If the quasi-neutral assumption is made, the Poisson equation is replaced by the constraint of zero local charge and the model reduces to the Isothermal Compressible Euler (ICE) model. We compare a numerical strategy based on the EPB model to a strategy using a reformulation (called REPB formulation). The REPB scheme captures the quasi-neutral limit more accurately

A high resolution wave propagation scheme for ideal two-fluid plasma equations

Abstract Algorithms for the solution of the five-moment ideal Two-Fluid equations are presented. The ideal Two-Fluid model is more general than the often used magnetohydrodynamic (MHD) model. The model takes into account electron inertia effects, charge separation and the full electromagnetic field equations and allows for separate electron and ion motion. The algorithm presented is the high resolution wave propagation method. The wave propagation method is based on solutions to the Riemann problem at cell interfaces. Operator splitting is used to incorporate the Lorentz and electromagnetic source terms. To preserve the divergence constraints on the electric and magnetic fields two different approaches are used. In the first approach Maxwell equations are rewritten in their mixed-potential form. In the second approach the so-called perfectly hyperbolic form of Maxwell equations are used which explicitly incorporate the divergence equations into the time stepping scheme. The algorithm is applied to a one-dimensional Riemann problem, ion-acoustic soliton propagation and magnetic reconnection. In each case Two-Fluid physics described by the ideal Two-Fluid model is highlighted

Sheared Flow As A Stabilizing Mechanism In Astrophysical Jets

It has been hypothesized that the sustained narrowness observed in the asymptotic cylindrical region of bipolar outflows from Young Stellar Objects (YSO) indicates that these jets are magnetically collimated. The j cross B force observed in z-pinch plasmas is a possible explanation for these observations. However, z-pinch plasmas are subject to current driven instabilities (CDI). The interest in using z-pinches for controlled nuclear fusion has lead to an extensive theory of the stability of magnetically confined plasmas. Analytical, numerical, and experimental evidence from this field suggest that sheared flow in magnetized plasmas can reduce the growth rates of the sausage and kink instabilities. Here we propose the hypothesis that sheared helical flow can exert a similar stabilizing influence on CDI in YSO jets.Comment: 13 pages, 2 figure

Evolution of Plasma Flow Shear and Stability in the ZaP Flow Z-Pinch

Abstract. The stabilizing effect of an axial flow on the m = 1 kink instability in Z-pinches has been studied numerically with a linearized ideal MHD model to reveal that a sheared axial flow stabilizes the kink mode when the shear exceeds a threshold. The sheared flow stabilizing effect is investigated with the ZaP Flow Zpinch experiment. An azimuthal array of surface mounted magnetic probes measures the fluctuation levels of the azimuthal modes m = 1, 2, and 3. After pinch assembly a quiescent period is found where the mode activity is significantly reduced. The quiescent period lasts for over 2000 times the expected instability growth time in a static Z-pinch. Optical images from a fast framing camera, a two-chord HeNe interferometer, and a ruby holographic interferometer indicate a stable, discrete pinch plasma during this time. Multichord Doppler shift measurements of impurity lines show a large, sheared flow during the quiescent period and low, uniform flow profiles during periods of high mode activity. The value of the velocity shear satisfies the theoretical threshold for stability during the quiescent period and does not satisfy the threshold during the high mode activity. Experiments are conducted with varying amounts of injected neutral gas to gain an understanding of the Z-pinch formation and lifetime

Entropy Stable Numerical Schemes for Two-Fluid Plasma Equations

Two-fluid ideal plasma equations are a generalized form of the ideal MHD equations in which electrons and ions are considered as separate species. The design of efficient numerical schemes for the these equations is complicated on account of their non-linear nature and the presence of stiff source terms, especially for high charge to mass ratios and for low Larmor radii. In this article, we design entropy stable finite difference schemes for the two-fluid equations by combining entropy conservative fluxes and suitable numerical diffusion operators. Furthermore, to overcome the time step restrictions imposed by the stiff source terms, we devise time-stepping routines based on implicit-explicit (IMEX)-Runge Kutta (RK) schemes. The special structure of the two-fluid plasma equations is exploited by us to design IMEX schemes in which only local (in each cell) linear equations need to be solved at each time step. Benchmark numerical experiments are presented to illustrate the robustness and accuracy of these schemes.Comment: Accepted in Journal of Scientific Computin

Pressure-driven instabilities in astrophysical jets

Astrophysical jets are widely believed to be self-collimated by the hoop-stress due to the azimuthal component of their magnetic field. However this implies that the magnetic field is largely dominated by its azimuthal component in the outer jet region. In the fusion context, it is well-known that such configurations are highly unstable in static columns, leading to plasma disruption. It has long been pointed out that a similar outcome may follow for MHD jets, and the reasons preventing disruption are still not elucidated, although some progress has been accomplished in the recent years. In these notes, I review the present status of this open problem for pressure-driven instabilities, one of the two major sources of ideal MHD instability in static columns (the other one being current-driven instabilities). I first discuss in a heuristic way the origin of these instabilities. Magnetic resonances and magnetic shear are introduced, and their role in pressure-driven instabilities discussed in relation to Suydam's criterion. A dispersion relation is derived for pressure-driven modes in the limit of large azimuthal magnetic fields, which gives back the two criteria derived by Kadomtsev for this instability. The growth rates of these instabilities are expected to be short in comparison with the jet propagation time. What is known about the potential stabilizing role of the axial velocity of jets is then reviewed. In particular, a nonlinear stabilization mechanism recently identified in the fusion literature is discussed. Key words: Ideal MHD: stability, pressure-driven modes; Jets: stabilityComment: 20 pages, 3 figures. Lecture given at the JETSET European school "Numerical MHD and Instabilities". To be published by Springer in the "Lectures notes in physics" serie

A general nonlinear fluid model for reacting plasma-neutral mixtures

A generalized, computationally tractable fluid model for capturing the effects of neutral particles in plasmas is derived. The model derivation begins with Boltzmann equations for singly charged ions, electrons, and a single neutral species. Electron-impact ionization, radiative recombination, and resonant charge exchange reactions are included. Moments of the reaction collision terms are detailed. Moments of the Boltzmann equations for electron, ion, and neutral species are combined to yield a two-component plasma-neutral fluid model. Separate density, momentum, and energy equations, each including reaction transfer terms, are produced for the plasma and neutral equations. The required closures for the plasma-neutral model are discussed