arXiv subjects

D. W. Crews

Publications and source records attributed to D. W. Crews.

6 recordsLinked to original sources

arXiv2025

Effects of transitional orbit magnetization on transport and current in Z pinches

The azimuthal self-magnetic field of the ideal Z pinch contains a central magnetic null. Trajectories around this null govern transport in the core. Particles follow cyclotron orbits when the guiding-center approximation holds. Approaching the field null, where the ordinary guiding-center regime breaks down, particles exhibit trajectories called, in some historical contexts, betatron orbits. We quantify transitional magnetization between cyclotron and betatron orbits by a magnetization parameter that decomposes phase space into these orbit regimes. Considering the distribution of all orbits, this phase-space decomposition reveals a transitional magnetization region wherein both populations coexist. Classical magnetized transport theory fails within this region, where the diamagnetic drift reverses. The drift flux is instead supported by the flux of betatron orbits. Kinematic diffusivity remains approximately constant rather than diverging at the null. These transport modifications are governed solely by the number density per unit length in the ideal pinch.

physics.plasm-ph↗

arXiv2024

Z Pinch Kinetics II -- A Continuum Perspective: Betatron Heating and Self-Generation of Sheared Flows

Adiabatic compression of a self-magnetizing current filament (a Z pinch) is analyzed via the adiabatic invariants of its constituent cyclotron and betatron motions. Chew-Goldberger-Low (CGL) models are recovered for both trajectories but with distinct anisotropy axes, about the magnetic field for cyclotron fluid and about the electric current for betatron fluid. In particular, betatron heating produces agyrotropic anisotropy which balances with gyrophase mixing. A hybrid CGL model is proposed based on the local densities of cyclotron and betatron orbits, then validated by numerical experiments. The relation between anisotropy and shear is explored by constructing the kinetic equilibrium of a flow expanded in the flux function. Flow as a linear flux function is simply bi-Maxwellian, while higher powers display higher-moment deviations. Next, weakly collisional gyroviscosity (magnetized pressure-strain) is considered in a forward process (forced flow) and an inverse process (forced anisotropy). The forward process phase-mixes flow into a simple flux function, freezing flow into flux and inducing anisotropy. In the inverse process, betatron heating-induced anisotropy self-generates a sheared flow to resist changes in flux. This flow, arising from momentum diffusion, is concentrated in the betatron region.

physics.plasm-ph↗

arXiv2024

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.

physics.plasm-ph↗

arXiv2022

Analysis of tensor-product discontinous Galerkin operators for Vlasov-Poisson simulations and GPU implementation on Python

The discontinuous Galerkin (DG) finite element method is conservative, lends itself well to parallelization, and is high-order accurate due to its close affinity with the theory of quadrature and orthogonal polynomials. When applied with an orthogonal discretization (\textit{i.e.} a rectilinear grid) the DG method may be efficiently implemented on a GPU in just a few lines of high-level language such as Python. This work demonstrates such an implementation by writing the DG semi-discrete equation in a tensor-product form and then computing the products using open source GPU libraries. The results are illustrated by simulating a problem in plasma physics, namely an instability in the magnetized Vlasov-Poisson system. Further, as DG is closely related to spectral methods through its orthogonal basis it is possible to calculate a transformation to an alternative set of global eigenfunctions for purposes of analysis or to perform additional operations. This transformation is also posed as a tensor product and may be GPU-accelerated. In this work a Fourier series is computed for example (although this does not beat discrete Fourier transform), and is used to solve the Poisson part of the Vlasov-Poisson system to $\mathcal{O}(Δx^{n+1/2})$-accuracy.

physics.comp-ph↗

arXiv2022

Arbitrarily high order implicit ODE integration by correcting a neural network approximation with Newton's method

As a method of universal approximation deep neural networks (DNNs) are capable of finding approximate solutions to problems posed with little more constraints than a suitably-posed mathematical system and an objective function. Consequently, DNNs have considerably more flexibility in applications than classical numerical methods. On the other hand they offer an uncontrolled approximation to the sought-after mathematical solution. This suggests that hybridization of classical numerical methods with DNN-based approximations may be a desirable approach. In this work a DNN-based approximator inspired by the physics-informed neural networks (PINNs) methodology is used to provide an initial guess to a Newton's method iteration of a very-high order implicit Runge-Kutta (IRK) integration of a nonlinear system of ODEs, namely the Lorenz system. In the usual approach many explicit timesteps are needed to provide a guess to the implicit system's nonlinear solver, requiring enough work to make the IRK method infeasible. The DNN-based approach described in this work enables large implicit time-steps to be taken to any desired degree of accuracy for as much effort as it takes to converge the DNN solution to within a few percent accuracy. This work also develops a general formula for the matrix elements of the IRK method for an arbitrary quadrature order.

math.NA↗

arXiv2022

On the validity of quasilinear theory applied to the electron bump-on-tail instability

The accuracy of quasilinear theory applied to the electron bump-on-tail instability, a classic model problem, is explored with conservative high-order discontinuous Galerkin methods applied to both the quasilinear equations and to a direct simulation of the Vlasov-Poisson equations. The initial condition is chosen in the regime of beam parameters for which quasilinear theory should be applicable. Quasilinear diffusion is initially in good agreement with the direct simulation but later underestimates the turbulent momentum flux. The direct simulation corrects from quasilinear evolution by quenching the instability in a finite time and producing a robust state of oscillation. Flux enhancement above quasilinear levels occurs as the phase space eddy turnover time in the largest amplitude wavepackets becomes comparable to the transit time of resonant phase fluid through wavepacket potentials. In this regime eddies effectively turn over during wavepacket transit so that phase fluid predominantly disperses by eddy phase mixing rather than by randomly phased waves. The enhanced turbulent flux of resonant phase fluid leads in turn, through energy conservation, to an increase in non-resonant turbulent flux and thus to an enhanced heating of the main thermal body above quasilinear predictions. These findings shed light on the kinetic turbulence fluctuation spectrum and support the theory that collisionless momentum diffusion beyond the quasilinear approximation can be understood through the dynamics of phase space eddies (or clumps and granulations).

physics.plasm-ph↗