Session VO04: Fundamental Plasmas: Computational and Analytical Techniques

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In this work, the dynamics of chaotic quantum maps is explored via simulation as a means to test the fidelity of emerging quantum computing hardware. Quantum computers promise to deliver enormous gains in computational power that can potentially be used to benefit Fusion Energy Sciences (FES). Through the quantum-classical correspondence principle, quantum systems of sufficiently large quantum number (or number of qubits) can approximate classical dynamics. Here we study the simplest types of chaotic dynamical systems, defined by classical and quantum maps. It's been shown that quantum maps with simple noise models can recreate small-scale classical phase space structures in the limit of many qubits [G. Benenti, et al. Phys. Rev. Lett. \textbf{87}, 227901-1 (2001)]. They can also deviate from the classical dynamics and display dynamical Anderson localization. A key next question is whether phase space structures can be clearly observed with current hardware, such as the LLNL Quantum Design and Integration Testbed (QuDIT) quantum computing platform. This is examined via simulations with experimentally derived noise models for the QuDIT platform.   

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Qualitative analysis techniques compliment computational ones in studying basic plasma systems. A Langmuir probe, immersed in a thermionically produced plasma source, measures plasma parameters yielding the so-called plasma characteristics I-V trace. By direct inspection, this trace resembles the logistic model curve developed before by Verhulst to account for population growth in relation to its sustaining resources. When a time-sweep of the Langmuir probe bias voltage is applied to the probe tip, plasma charge current is collected between the two probe bias voltage polarities. The charge content of the plasma and its sustaining energy, as sensed by the probe bias voltage time-sweep, can be measured. Moreover, and at the probe current saturation region, the total charge collected by the probe can be measured using the law of mass action. This is done without imposing any additional assumptions as to whether the charges are discrete (carried by electrons) or continuous (as in the space charge effect). Both the transitive mixed states of charges, as exhibited in the Langmuir probe trace, and the computational model, used in this paper, reveal the plasma system chaotic behaviour.   

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The Rayleigh- Taylor (RT) instability occurs along the interface between two fluids, when a layer of lighter fluid is pushing upon a denser one, in the presence of a gravitational field or external potential. In convergent geometry, such as spherical or cylindrical configurations, RT growth is modified by Bell-Plesset (BP) effects. We investigate various linear and nonlinear models for RT and BP growth in convergent geometry, including those that allow for compressibility of the fluids. In particular, we focus on the explicit characterization of BP effects in the underdriven or 'accelerationless' limit where the contribution from RT growth is small. The models are applied to recent cylindrical implosion experiments that directly measure hydrodynamic instability growth in convergent geometries. Extensions of the models are considered, and we apply bifurcation analysis in this setting to study the evolution of our system under parameter variation.   

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Geometrical optics (GO) is often used to model wave propagation in weakly inhomogeneous media. However, GO predicts spurious singularities of the wave field near reflection points and, more generally, caustics. This is problematic for applications such as using GO codes to optimize plasma heating and current drive by radiofrequency waves. We present a new formulation of GO, called metaplectic geometrical optics (MGO), that is free from wave field singularities at caustics and thus can be used to develop more versatile codes. We then present examples of how MGO can be used to accurately describe electromagnetic plasma waves propagating in various density profiles.   

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We present a new forward semi-Lagrangian particle method for the Vlasov-Poisson (VP) system. Recent methods for solving the VP system include deformable particles and high-order and/or discontinuous-Galerkin Eulerian methods. In contrast to these, we do not use any operator splitting and obtain the electric field by summing regularized pairwise particle interactions using a hierarchical treecode. We use remeshing and adaptive mesh refinement to maintain an efficient representation of phase space. We benchmark on several standard test cases including Landau damping and the two-stream instability.   

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The Boltzmann equation forms the foundation of almost all of plasma physics, as well as other subfields of physics and engineering. Recent advances in applied mathematics have made routine direct numerical solution well within reach. A conservative spectral method is being applied to obtain more accurate coupling between high fidelity gyrokinetic and neutral simulation models. In addition, a standalone solver for more general application has been developed. Recent advances include the addition of spatial dependence, field acceleration, and implicit time-stepping. The algorithm has been benchmarked against a manufactured solution for Maxwell molecules, and against the Chapman-Enskog fluid expansion at low Knudsen number. Results for electron swarm parameters in weakly ionized plasmas are compared with other approaches to solving the Boltzmann equation. One-dimensional simulation of neutrals under detachment conditions is also demonstrated as a proof-of-principle. The precomputed discrete collision operators allow all these cases to be run locally on a workstation or laptop, even with the nonlinear collision operator.   

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We report on studies of the Buneman type instability driven by the relative drift of warm electrons with respect to warm ions. A series of highly resolved PIC and Vlasov simulations are performed and compared. Linear simulations show that although the results of PIC simulations from several different codes agree, they are different from Vlasov simulations when the beam electron velocity is relatively low. The difference between the PIC and Vlasov results is explained by the electron trapping in initial noisy distributions of the electric potential. In the deeply nonlinear stage, we observe strong modification of the electron distribution function and generation of backward waves (in the direction opposite to the velocity of the electron beam). The backward waves are the result of the wave-particle and resonant mode excitation by the electron beams from the modified electron distribution function. The formation and evolution of the electron and ions holes are further investigated.   

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We present a novel Particle-in-Cell (PIC) simulation scheme on unstructured meshes for studying low-frequency electrostatic perturbations in magnetized plasmas. In this scheme, ions are treated as fully kinetic~(6-dimensional)~particles, and electrons are described by the adiabatic response. This PIC scheme is derived from a discrete variational principle [1-3] for electrostatic perturbations on unstructured meshes. To preserve the geometric structure of the system, the discrete variational principle requires that on an unstructured mesh charge is deposited with Whitney 0-forms and the electric field is interpolated using Whitney 1-forms. The new PIC scheme has been implemented on a 2-D triangular unstructured mesh and applied to study Ion Bernstein Waves (IBW). The IBW simulation results agree well with the analytic dispersion relation [4]. The implementation of the algorithm on 3-D unstructured mesh will also be discussed. [1] Squire, Qin and Tang, PoP 19, 08451 (2012). [2] Xiao, Qin et al, PoP 22, 112504 (2015). [3] Xiao and Qin, NF 59,~106044~(2019). [4]~Sturdevant, Benjamin, \textit{PhD Dissertation} (2016).   

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Traditional Boltzmann kinetic theory models the Coulomb collisions of unmagnetized and weakly magnetized plasmas in which the typical gyroradius is larger than the Debye length. Conversely, O'Neil's kinetic theory models Coulomb collisions of extremely magnetized plasma transport regime in which the typical gyroradius is smaller than the distance of closest approach. Here, we develop a generalized collision operator that can treat Coulomb collisions in plasmas across all magnetization strength regimes and which asymptotes to the traditional kinetic theory, or O'Neil's theory, in the appropriate limits. To demonstrate the utility of the collision operator, it is used to compute the friction force on a massive test particle. In the strong magnetization regimes, the friction force is found to have a transverse component that is perpendicular to both velocity and Lorentz force in addition to the stopping power as predicted by linear response theory. Good agreement is found between the collision theory and the linear response theory in the regime in which both apply, but the generalized collision operator extends the theory to regimes inaccessible using the linear response theory.   

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We derive a Noether current for the Eulerian variational principle of ideal non-barotropic magnetohydrodynamics (MHD). It was shown previously that ideal non-barotropic MHD is mathematically equivalent to a five function field theory with an induced geometrical structure in the case that field lines cover surfaces and this theory can be described using a variational principle. Here we use various symmetries of the flow to derive novel topological constants of motion through the newly derived Noether current and discuss their implication for non-barotropic MHD and plasma confinement.   

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We consider weakly ionized plasma where the frequent elastic scattering of electrons on neutrals changes the character of electron-electron collisions entirely. This comes into play when the frequency of the electron -- neutral collisions is so large that the corresponding electron mean-free path is shorter than the closest distance between the colliding electrons. In this extreme case, the electron energy equilibration differs considerably from that in fully ionized plasma. A crucial role is now played by diffusion of electrons caused by their scattering on neutrals, and we demonstrate how a proper account of this diffusion allows one to estimate the characteristic energy equilibration time for electrons. We also present a rigorous derivation of the kinetic equation for electrons via Bogolyubov's approach based on Liouville equations for multi-particle distribution functions.   

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Recently, frequency of the MHD modes are modulated using periodic gas-puffs in the ADITYA-U tokamak have been investigated [1,2]. Further, experiments using biased electrodes have previously shown disruption avoidance by controlling the MHD modes effectively in the ADITYA tokamak [3]. In this presentation, we have attempted to simulate the MHD activities in the ADITYA/ADITYA-U tokamak using an MHD code [4]. Particularly, the effect of flows on MHD modes are studied using the above-mentioned code and the results are compared with experimental observations. The simulation results of effect of gas-puffing and electrode biasing on MHD modes in ADITYA/ADITYA-U tokamak are presented in this paper. References: [1] Overview of recent experimental results from the Aditya tokamak, R. Tanna et al. October 2017, Nuclear Fusion 57(10):102008 [2] Effect of periodic gas-puffs on drift-tearing modes in ADITYA/ADITYA-U tokamak discharges, Nuclear Fusion, Volume 60, Number 3, 036012 [2] A novel approach for mitigating disruptions using biased electrode in Aditya tokamak. Nucl. Fusion 54 (2014) 083023 [4] Visco-resistive MHD study of internal kink (m $=$ 1) modes, J. Mendonca et al, Physics of Plasmas 25, 022504 (2018)   

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In 1601, Kepler inherited the observational data of planetary orbits meticulously collected by Tycho Brahe. It took Kepler 5 years to discover his laws of planetary motion, and another 78 years for Newton to solve the Kepler problem using his laws of motion and gravitation. Recently I developed a machine learning and serving algorithm for discrete field theories that applies to a wide range of physics problems [arXiv:1910.10147]. The algorithm learns a discrete field theory from observational data and then directly predicts new observations without laws of physics. In particular, the algorithm solves the Kepler problem without learning or knowing Newton's laws of motion and gravitation. The learning algorithm learns a discrete field theory from a set of planetary orbit data similar to what Kepler inherited, and the serving algorithm correctly predicts other planetary orbits, including parabolic and hyperbolic escaping orbits, of the solar system. The proposed algorithm is also applicable when relativistic effects are important without knowing or learning Einstein's theory. The illustrated advantages of discrete field theories for machine learning are consistent with Bostrom's simulation hypothesis. I will also show how this algorithm can help to achieve fusion energy.