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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

A Particle-in-Cell Method for Plasmas with a Generalized Momentum Formulation, Part II: Enforcing the Lorenz Gauge Condition

In a previous paper Christlieb et al. (A particle-in-cell method for plasmas with a generalized momentum formulation, part I: Model formulation, 2024), we developed a new particle-in-cell (PIC) method for the relativistic Vlasov–Maxwell system in which the electromagnetic fields and the equations of motion for the particles were cast in terms of scalar and vector potentials through a Hamiltonian formulation. This new method evolved the potentials under the Lorenz gauge using integral equation methods. New methods to construct spatial derivatives of the potentials that converge at the same rates as the fields were also presented. The new particle method was compared against standard explicit discretizations, including the well-known FDTD-PIC method, for a range of applications involving sheaths and particle beams. Here, this paper extends this new class of methods by focusing on the enforcement the Lorenz gauge condition in both exact and approximate forms using co-located meshes. A time-consistency property of the proposed field solver for the vector potential form of Maxwell’s equations is established, which is shown to preserve the equivalence between the semi-discrete Lorenz gauge condition and the analogous semi-discrete continuity equation. Using this property, we present three methods to enforce a semi-discrete gauge condition. The first method introduces an update for the continuity equation that is consistent with the discretization of the Lorenz gauge condition. Both the finite difference and spectral implementations satisfy this discrete gauge condition to machine precision. The second approach we propose enforces a semi-discrete continuity equation using the boundary integral solution to the field equations. The potential benefit of this approach is that it eliminates spatial derivatives that appear on the particle data, namely the current density, which is often calculated by linear combinations of low-order spline basis functions. This method is ideally suited to boundary integral equation methods that invert multi-dimensional operators without dimensional splitting techniques and will be the subject of future work. The third approach introduces a gauge correcting method that makes direct use of the gauge condition to modify the scalar potential and uses local maps for both the charge and current densities. This results in a gauge error, as the maps do not enforce the continuity equation. The vector potential coming from the current density is taken to be exact, and using the Lorenz gauge, we compute a correction to the scalar potential that makes the two potentials satisfy the gauge condition. This method also enforces the gauge condition to machine precision. We demonstrate two of the proposed methods in the context of periodic domains. Problems defined on bounded domains, including those with complex geometric features remain an ongoing effort. However, this work shows that it is possible to design computationally efficient methods that can effectively enforce the Lorenz gauge condition in a non-staggered PIC formulation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A Particle-in-Cell Method for Plasmas with a Generalized Momentum Formulation, Part I: Model Formulation

Here, this paper formulates a new particle-in-cell method for the Vlasov–Maxwell system. Under the Lorenz gauge condition, Maxwell’s equations for the electromagnetic fields can be written as a collection of scalar and vector wave equations. The use of potentials for the fields motivates the adoption of a Hamiltonian formulation for particles that employs the generalized (conjugate) momentum. A notable advantage offered by the Hamiltonian formulation is the elimination of time derivatives in the Lorenz gauge formulation that are required by the standard Newton–Lorentz treatment of the particles. This allows the fields to retain the full time-accuracy guaranteed by the field solver. The resulting updates for particles require only knowledge of the fields and their spatial derivatives. An analytical method for constructing these spatial derivatives is presented that exploits the underlying integral solution used in the field solver for the wave equations. Moreover, these derivatives are demonstrated to converge at the same rate as the fields in both time and space. The Method of Lines Transpose field solver we consider in this work is globally first-order accurate in time and high-order accurate in space (e.g., fourth- and fifth-order) and belongs to a larger class of methods which are unconditionally stable, can address geometry, and leverage $\mathcal {O}(N)$ fast summation methods for efficiency. We demonstrate the method on several well-established benchmark problems on bounded domains, including a plasma sheath as well as a relativistic particle beam. The efficacy of the proposed formulation is established by comparing with a second-order accurate finite-difference time-domain method that employs a leapfrog time advance for particles and a charge conserving map suitable for bounded domains. The new method shows mesh-independent numerical heating properties even in cases where the plasma Debye length is smaller than the grid spacing. This is an important feature of the new method for problems defined on bounded domains, because it permits the use of coarser grids in space in the representation of the fields. Such a capability has significant implications for the simulation of plasmas in bounded domains with complex geometry, where the ratio between the largest and smallest cells can vary significantly. The use of high-order spatial approximations in the new method also means that fewer grid points are required in order to achieve a fixed accuracy. Our results also suggest that the new method can be used with fewer simulation particles per cell compared to the benchmark explicit method, which permits further computational savings.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2015

Generalized Momentum Control of the Spin-Stabilized Magnetospheric Multiscale Formation

The Magnetospheric Multiscale (MMS) mission consists of four identically instrumented, spin-stabilized observatories elliptically orbiting the Earth in a tetrahedron formation. The on-board attitude control system adjusts the angular momentum of the system using a generalized thruster-actuated control system that simultaneously manages precession, nutation and spin. Originally developed using Lyapunov control-theory with rate-feedback, a published algorithm has been augmented to provide a balanced attitude/rate response using a single weighting parameter. This approach overcomes an orientation sign-ambiguity in the existing formulation, and also allows for a smoothly tuned-response applicable to both a compact/agile spacecraft, as well as one with large articulating appendages.

Maneuver and Attitude Dynamics↗
DOE OSTI2025

A Particle-in-cell Method for Plasmas with A Generalized Momentum Formulation, Part III: A family of Gauge Conserving Methods

In this paper, we introduce a new family of spatially co-located field solvers for particle-in-cell applications which evolve the potential formulation of Maxwell’s equations under the Lorenz gauge. Our recent work [2] introduced the concept of time-consistency, which connects charge conservation to the preservation of the gauge at the semi-discrete level. It will be shown that there exists a large family of time discretizations which satisfy this property. Additionally, it will be further shown that for large classes of time marching methods, the satisfaction of the gauge condition automatically implies the satisfaction of Gauss’s law for electricity, with the potential formulation ensuring that that Gauss’s law for magnetism is satisfied by definition. We focus on popular time marching methods including centered differences, backward differences, and diagonally-implicit Runge-Kutta methods, which are coupled to a spectral discretization in space. We demonstrate the theory by testing the methods on a relativistic Weibel instability and a drifting cloud of electrons.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2015

Generalized Momentum Control of the Spin-Stabilized Magnetospheric Multiscale (MMS) Formation

Angular momentum control maneuvers required to keep spin-axis in science box. Traditional approach uses de-coupled modes for pointing, spin, nutation Impractical for MMS Frequency and Number of maneuvers (Orbit Control, Pointing, Nutation, Spin, four observatories, every 2-4 weeks). Difficult to implement de-coupled open-loop control with flexible wire booms. Desire a unified angular momentum controller. Comprehensively control pointing, spin, and nutation.

Maneuver and Attitude Dynamics↗
NASA NTRS1973

Normal and Tangential Momentum Accommodation for Earth Satellite Conditions

Momentum accommodation was determined experimentally for gas-surface interactions simulating in a practical way those of near-earth satellites. Throughout the ranges of gas energies and incidence angles of interest for earth-conditions, two components of force were measured by means of a vacuum microbalance to determine the normal and tangential momentum-accommodation coefficients for nitrogen ions on technical-quality aluminum surfaces. For these experimental conditions, the electrodynamics of ion neutralization near the surface indicate that results for nitrogen ions should differ relatively little from those for nitrogen molecules, which comprise the largest component of momentum flux for near-earth satellites. The experimental results indicated that both normal and tangential momentum-accommodation coefficients varied widely with energy, tending to be relatively well accommodated at the higher energies, but becoming progressively less accommodated as the energy was reduced to and below that for earth-satellite speeds. Both coefficients also varied greatly with incidence angle, the normal momentum becoming less accommodated as the incidence angle became more glancing, whereas the tangential momentum generally became more fully accommodated. For each momentum coefficient, an empirical correlation function was obtained which closely approximated the experimental results over the ranges of energy and incidence angle. Most of the observed variations of momentum accommodation with energy and incidence angle were qualitatively indicated by a calculation using a three-dimensional model that simulated the target surface by a one-dimensional attractive potential and hard sphere reflectors.

Knechtel, Earl D.↗
NASA NTRS1957

A Theoretical Investigation of the Drag of Generalized Aircraft Configurations in Supersonic Flow

It seems possible that, in supersonic flight, unconventional arrangements of wings and bodies may offer advantages in the form of drag reduction. It is the purpose of this report to consider the methods for determining the pressure drag for such unconventional configurations, and to consider a few of the possibilities for drag reduction in highly idealized aircraft. The idealized aircraft are defined by distributions of lift and volume in three-dimensional space, and Hayes' method of drag evaluation, which is well adapted to such problems, is the fundamental tool employed. Other methods of drag evaluation are considered also wherever they appear to offer amplifications. The basic singularities such as sources, dipoles, lifting elements and volume elements are discussed, and some of the useful inter-relations between these elements are presented. Hayes' method of drag evaluation is derived in detail starting with the general momentum theorem. In going from planar systems to spatial systems certain new problems arise. For example, interference between lift and thickness distributions generally appears, and such effects are used to explain the difference between the non-zero wave drag of Sears-Haack bodies and the zero wave drag of Ferrari's ring wing plus central body. Another new feature of the spatial systems is that optimum configurations generally are not unique, there being an infinite family of lift or thickness distributions producing the same minimum drag. However it is shown that all members of an optimum family produce the same flow field in a certain region external to the singularity distribution. Other results of the study indicate that certain spatial distributions may produce materially less wave drag and vortex drag than comparable planar systems. It is not at all certain that such advantages can be realized in practical aircraft designs, but further investigation seems to be warranted.

Graham, E. W.↗
NASA NTRS1974

Three-body Coulomb systems using generalized angular-momentum S states

An expansion of the three-body Coulomb potential in generalized angular-momentum eigenfunctions developed earlier by one of the authors is used to compute energy eigenvalues and eigenfunctions of bound S states of three-body Coulomb systems. The results for He, H(-), e(-)e(+)e(-), and pmu(-)p are compared with the results of other computational approaches.

Whitten, R. C.↗
NASA NTRS1972

Optimum performance of hovering rotors

A theory for the optimum performance of a rotor hovering out of ground effect is developed. The performance problem is formulated using general momentum theory for an infinitely bladed rotor, and the effect of a finite number of blades is estimated. The analysis takes advantage of the fact that a simple relation exists between the radial distributions of static pressure and angular velocity in the ultimate wake, far downstream of the rotor, since the radial velocity vanishes there. This relation permits the establishment of an optimum performance criterion in terms of the ultimate wake velocities by introducing a small local perturbation of the rotational velocity and requiring the resulting ratio of thrust and power changes to be independent of the radial location of the perturbation. This analysis fully accounts for the changes in static pressure distribution and axial velocity distribution throughout the wake as the result of the local perturbation of the rotational velocity component.

Wu, J. C.↗
NASA NTRS1987

Development of a two-dimensional zonally averaged statistical-dynamical model. II - The role of eddy momentum fluxes in the general circulation and their parameterization

The role of eddy momentum fluxes in the general circulation was investigated using a two-dimensional zonally averaged statistical-dynamical model described by Yao and Stone (1987), which is almost two orders of magnitude faster than the three-dimensional climate model of Hansen et al. (1983). Results show that the vertical structure of the meridional eddy flux has relatively little impact on the general circulation, presumably because the vertical structure is strongly constrained by the thermal wind relation and surface friction. On the other hand, it was found that, in order to simulate accurately the general circulation and its response to climate changes, parameterization of the vertically integrated meridional eddy flux of angular momentum is necessary. A new parameterization of this eddy momentum transport was carried out, which is intended to represent the transport due to large-scale transient eddies arising from baroclinic instability.

Stone, Peter H.↗
NASA NTRS1974

The effect of small-scale vertical mixing of horizontal momentum in a general circulation model

Several experiments are described in which the sub-grid-scale vertical eddy viscosity in the GISS global general circulation model was varied. The results show that large viscosities suppress large-scale eddies in middle and high latitudes, but enhance the circulation in the tropical Hadley cell and increase the extent of the tropical easterlies. Comparison with observations shows that the GISS model requires eddy viscosities about 1 sq m per sec or less to give realistic results for middle and high latitudes, and eddy viscosities about 100 sq m per sec to give realistic results for low latitudes. A plausible mechanism for the implied increase in small-scale mixing in low latitudes is cumulus convection.

Stone, P. H.↗
NASA NTRS1992

Spin-up of a rapidly rotating star by angular momentum loss - Effects of general relativity

It has recently been shown that a rapidly rotating Newtonian star can spin up by radiating angular momentum. Extremely fast pulsars losing energy and angular momentum by magnetic dipole radiation or gravitational radiation may exhibit this behavior. Here, we show that this phenomenon is more widespread for rapidly rotating stars in general relativity. We construct and tabulate polytropic sequences of fully relativistic rotating stars of constant rest mass and entropy. We find that the range of adiabatic indices allowing spin-up extends somewhat above 4/3 because of the nonlinear effects of relativistic gravity. In addition, there is a new class of 'supramassive' stars which will inevitably spin up by losing angular momentum regardless of their equation of state. A supramassive star, spinning up via angular momentum loss, will ultimately evolve until it becomes unstable to catastrophic collapse to a black hole. Spin-up in a rapidly rotating star may thus be an observational precursor to such collapse.

Cook, Gregory B.↗
DOE OSTI2026

Boundary-induced classical generalized Gibbs ensemble with angular momentum

We investigate how confinement geometry leads to the emergence of a Generalized Gibbs Ensemble (GGE) in classical systems. Unlike the standard Gibbs ensemble, the GGE includes additional conserved quantities, such as angular momentum, that arise from boundary-induced symmetries. Using analytical arguments based on the maximum entropy principle, we show that circular boundaries preserve angular momentum and drive the system toward a chiral, non-ergodic GGE that violates time-reversal symmetry. This ensemble differs fundamentally from the Gibbs case, producing near-boundary condensation and revealing how geometry alone can alter thermal equilibration. To quantify these effects, we introduce an order parameter measuring deviations from Gibbs behavior and demonstrate that conventional Monte Carlo methods must incorporate angular momentum conservation under such conditions. Our study highlights how geometric constraints shape non-equilibrium statistical ensembles and lead to subtle departures from the Bohr-van Leeuwen theorem. These predictions are validated through detailed simulations of confined classical hard-disk gases.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1984

An analysis of the high-latitude thermospheric wind pattern calculated by a thermospheric general circulation model. I - Momentum forcing

A diagnostic processor (DP) was developed for analysis of hydrodynamic and thermodynamic processes predicted by the NCAR thermospheric general circulation model (TGCM). The TGCM contains a history file on the projected wind, temperature and composition fields at each grid point for each hour of universal time. The DP assimilates the history file plus ion drag tensors and drift velocities, specific heats, coefficients of viscosity, and thermal conductivity and calculates the individual forcing terms for the momentum and energy equations for a given altitude. Sample momentum forcings were calculated for high latitudes in the presence of forcing by solar radiation and magnetospheric convection with a 60 kV cross-tail potential, i.e., conditions on Oct. 21, 1981. It was found that ion drag and pressure forces balance out at F region heights where ion drift velocities are small. The magnetic polar cap/auroral zone boundary featured the largest residual force or net acceleration. Diurnal oscillations were detected in the thermospheric convection, and geostrophic balance was dominant in the E layer.

Killeen, T. L.↗
NASA NTRS2004

Atmospheric Rotational Effects on Mars Based on the NASA Ames General Circulation Model: Angular Momentum Approach

The objective of the investigation is to determine the motion of the rotational axis of Mars as a result of mass variations in the atmosphere and condensation and sublimation of CO2 ice on the polar caps. A planet experiences this type of motion if it has an atmosphere, which is changing its mass distribution with respect to the solid body of the planet and/or it is asymmetrically changing the amount of ice at the polar caps. The physical principle involved is the conservation of angular momentum, one can get a feeling for it by sitting on a well oiled swivel chair holding a rotating wheel on a horizontal direction and then changing the rotation axis of the wheel to a vertical direction. The person holding the wheel and the chair would begin to rotate in opposite direction to the rotation of the wheel. The motions of Mars atmosphere and the ice caps variations are obtained from a mathematical model developed at the NASA Ames Research Center. The model produces outputs for a time span of one Martian year, which is equivalent to 687 Earth days. The results indicate that Mars axis of rotation moves in a spiral with respect to a reference point on the surface of the planet. It can move as far away as 35.3 cm from the initial location as a result of both mass variations in the atmosphere and asymmetric ice variations at the polar caps. Furthermore the pole performs close to two revolutions around the reference point during a Martian year. This motion is a combination of two motions, one produced by the atmospheric mass variations and another due to the variations in the ice caps. The motion due to the atmospheric variations is a spiral performing about two and a half revolutions around the reference point during which the pole can move as far as 40.9 cm. The motion due to variations in the ice caps is a spiral performing almost three revolutions during which the pole can move as far as 32.8 cm.

Sanchez, Braulio V.↗
DOE OSTI2026

Unpolarized GPDs at small x and non-zero skewness

We study the small-x asymptotics of unpolarized generalized parton distributions (GPDs) and generalized transverse momentum distributions (GTMDs). Unlike the previous works in the literature, we consider the case of non-zero (but small) skewness while allowing for non-linear contributions to the evolution equations. We first show that unpolarized GPDs and GTMDs at small x are related to the eikonal dipole amplitude N, whose small-x evolution is given by the BK/JIMWLK evolution equations, and to the odderon amplitude $\mathscr{O}$, whose evolution is also known in the literature. We then show that the effect of non-zero skewness ξ ≠ 0 is to modify the value of the evolution parameter (rapidity) in the arguments for the dipole amplitudes N and $\mathscr{O}$ from Y = ln(1/x) to Y = ln min{1/|x|, 1/|ξ|}.

Generalized parton distributions (GPDs)↗