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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

General formulas for adiabatic invariants in nearly periodic Hamiltonian systems

While it is well known that every nearly periodic Hamiltonian system possesses an adiabatic invariant, extant methods for computing terms in the adiabatic invariant series are inefficient. The most popular method involves the heavy intermediate calculation of a non-unique near-identity coordinate transformation, even though the adiabatic invariant itself is a uniquely defined scalar. Here, a less well-known method, developed by S. Omohundro, avoids calculating intermediate sequences of coordinate transformations but is also inefficient as it involves its own sequence of complex intermediate calculations. In order to improve the efficiency of future calculations of adiabatic invariants, we derive generally applicable, readily computable formulas for the first several terms in the adiabatic invariant series. To demonstrate the utility of these formulas, we apply them to charged-particle dynamics in a strong magnetic field and magnetic field-line dynamics when the field lines are nearly closed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Electron Drift Kinetics in Space and in the Lab [Slides]

In a magnetized plasma, dynamics are fundamentally split between the directions parallel and perpendicular to the magnetic field. Generally, gyration about the magnetic field will occur on smaller spatial and time scales than the bulk dynamics of the system. From Hamiltonian mechanics, a periodic motion on a time scale much shorter than the other system dynamics leads to an adiabatic invariant. The adiabatic invariant for this motion is the magnetic moment $\mu=\frac{mv⟂^2}{2B}$, which will be approximately conserved.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Energizing charged particles by an orbit instability in a slowly rotating magnetic field

In this work, the stability of charged particle motion in a uniform magnetic field with an added spatially uniform transverse rotating magnetic field (RMF) is studied analytically. It is found that the stability diagram of a single particle's orbit depends critically on the chosen boundary conditions. We show that for many boundary conditions and wide regions in the parameter space, RMFs oscillating far below the cyclotron frequency can cause linear instabilities in the motion which break μ invariance and energize particles. Such energization may appear at odds with the adiabatic invariance of μ; however, adiabatic invariance is an asymptotic result and does not preclude such heating by magnetic fields oscillating at slow frequencies. This mechanism may contribute to heating in the edge plasma of field-reversed configurations (FRCs) in rotamak-FRC experiments. Furthermore, these RMF-driven instabilities may significantly enhance azimuthal current drive during the formation of FRCs in such devices.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

Abstract M. Kruskal showed that each continuous-time nearly periodic dynamical system admits a formal U (1)-symmetry, generated by the so-called roto-rate. When the nearly periodic system is also Hamiltonian, Noether’s theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal’s theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along a U (1)-action. When the limiting rotation is non-resonant, these maps admit formal U (1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formal U (1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether’s theorem. When the unperturbed U (1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Derivations of the Total Radiation Belt Electron Content

Abstract We present multiple derivations of the Total Radiation Belt Electron Content (TRBEC), an indicator of the global number of electrons that instantaneously occupy the radiation belts. Derived from electron flux measurements, the TRBEC reduces the spatial information into a scalar quantity that concisely describes global aspects of the system. This index provides a simple, global, and long‐term assessment of the radiation belts that enables systematic analysis. In this work, we examine the TRBEC using the adiabatic invariants of which has been used in previous articles as this coordinate system removes reversible adiabatic effects. We then introduce a new expression to compute the TRBEC using the non‐adiabatic coordinates of , relevant in the contexts of energetic electron precipitation, chorus, and hiss scattering where adiabatic invariant quantities are no longer conserved. From both expressions of the TRBEC we demonstrate that an erroneous factor of that appeared in previous works using the adiabatic derivation led to an overestimate of the reported electron populations. In addition, we quantify electron loss in the outer radiation belt via a case study using the Van Allen Probes data over a 20‐day period from March 2013 specifying particle populations both in terms of the aforementioned adiabatic and non‐adiabatic variables. The total number of electrons in the outer radiation belt reached upwards of electrons at the peak of the storm, a rest mass of roughly 10 g.

Astronomy & Astrophysics↗
DOE OSTI2023

Approximation of nearly-periodic symplectic maps via structure-preserving neural networks

A continuous-time dynamical system with parameter ε is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as ε approaches 0. Nearly-periodic maps are discrete-time analogues of nearly-periodic systems, defined as parameter-dependent diffeomorphisms that limit to rotations along a circle action, and they admit formal U(1) symmetries to all orders when the limiting rotation is non-resonant. For Hamiltonian nearly-periodic maps on exact presymplectic manifolds, the formal U(1) symmetry gives rise to a discrete-time adiabatic invariant. In this paper, we construct a novel structure-preserving neural network to approximate nearly-periodic symplectic maps. This neural network architecture, which we call symplectic gyroceptron, ensures that the resulting surrogate map is nearly-periodic and symplectic, and that it gives rise to a discrete-time adiabatic invariant and a long-time stability. This new structure-preserving neural network provides a promising architecture for surrogate modeling of non-dissipative dynamical systems that automatically steps over short timescales without introducing spurious instabilities.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Pitch-Angle Diffusion in the Earth’s Magnetosphere Organized by the Mozer-Transformed Coordinate System

In the Earth’s dipole magnetosphere finite-gyroradius effects produce a shift of the atmospheric loss cone away from the direction of the magnetic field. This loss-cone shift is theoretically described by the “Mozer transform”, which is based upon the curvature drift of particles crossing the equatorial plane. For positive ions the northern and southern loss cones both shift westward and for electrons the northern and southern loss cones both shift eastward. This loss-cone shift is part of a coordinate-system transform, with the transformed coordinates better organizing the behavior of particle orbits in the dipole magnetic field (e.g. first adiabatic invariants, mirror heights, and bounce times). In this report it is demonstrated that the transformed coordinate system also properly organizes pitch-angle diffusion. This improved organization of the diffusion is true whether the angular scattering is produced by plasma-wave scattering or by field-line-curvature (FLC) scattering. It is shown that FLC scattering and the loss cone shift are linked, so that if FLC scattering is occurring, there is a loss cone shifted away from the magnetic-field direction and the Mozer-transformed coordinates are needed.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Modification of the Loss Cone for Energetic Particles in the Earth's Inner Magnetosphere

The behavior of energetic particles in the Earth's dipole and stretched magnetic-field models is explored with a focus on the shift of the atmospheric loss cone away from the magnetic-field direction and on non-adiabatic behavior occurring when the particle gyroradius becomes comparable with the gradient scale length of the local magnetic field. It is shown that the equatorial loss cone is aberrated away from the magnetic-field direction (pitch angle of 0°) by the perpendicular drift of a charged particle (which is referred to as the “Mozer transform” described in Mozer (1966, https://doi.org/10.1029/JZ071i011p02701). It is found that the Mozer coordinate transformation in pitch-angle/gyrophase-angle space better organizes the behavior of bounce times, mirror altitudes, drift speeds, and first adiabatic invariant. It also describes the loss-cone shift accurately for a certain range of values of the adiabaticity parameter ϵ, defined by the ratio of the particle gyroradius to the local radius of curvature of the magnetic field. For particles with larger ϵ, the Mozer transform (evaluated with the standard gradient-curvature drift) breaks down and the “central trajectory” theory can be used to calculate the angular shift of the loss cone, which now includes an Earthward shift. When ϵ becomes relatively large, stochastic field line curvature (FLC) scattering occurs: we show the intimate connection between the strength of FLC scattering and the loss-cone shift. Finally, by comparing ion orbits in the dipole and Tsyganenko (Ts89 and Ts04) magnetic fields, we conclude that the loss cone of ring-current ions is significantly modified during geomagnetically active times.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Normal stability of slow manifolds in nearly periodic Hamiltonian systems

Kruskal [J. Math. Phys. 3, 806 (1962)] showed that each nearly periodic dynamical system admits a formal U(1) symmetry, generated by the so-called roto-rate. We prove that such systems also admit nearly invariant manifolds of each order, near which rapid oscillations are suppressed. We study the nonlinear normal stability of these slow manifolds for nearly periodic Hamiltonian systems on barely symplectic manifolds—manifolds equipped with closed, non-degenerate 2-forms that may be degenerate to leading order. In particular, we establish a sufficient condition for long-term normal stability based on second derivatives of the well-known adiabatic invariant. We use these results to investigate the problem of embedding guiding center dynamics of a magnetized charged particle as a slow manifold in a nearly periodic system. Here, we prove that one previous embedding and two new embeddings enjoy long-term normal stability and thereby strengthen the theoretical justification for these models.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Axion dark matter from frictional misalignment

We study the impact of sphaleron-induced thermal friction on the axion dark-matter abundance due to the interaction of an axion-like particle (ALP) with a dark non-abelian gauge sector in a secluded thermal bath. Thermal friction can either enhance the axion relic density by delaying the onset of oscillations or suppress it by damping them. We derive an analytical formula for the frictional adiabatic invariant, which remains constant along the axion evolution and which allows us to compute the axion relic density in a general set-up. Even in the most minimal scenario, in which a single gauge group is responsible for both the generation of the ALP mass and the friction force, we find that the resulting dark-matter abundance from the misalignment mechanism deviates from the standard scenario for axion masses m a ≳ 100 eV. We also generalize our analysis to the case where the gauge field that induces friction and the gauge sector responsible for the ALP mass are distinct and their couplings to the axion have a large hierarchy as can be justified by means of alignment or clockwork scenarios. We find that it is easy to open up the ALP parameter space where the resulting axion abundance matches the observed dark-matter relic density both in the traditionally over- and underabundant regimes. This conclusion also holds for the QCD axion.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2026

Verification of nonperturbative guiding center theory in symmetric fields

We verify a recently-developed nonperturbative guiding center formalism to charged particle dynamics in fields with two-parameter continuous symmetry groups. This entails finding exact constants of motion, valid in the nonperturbative regime, that agree with Kruskal’s adiabatic invariant series to all orders in the perturbative regime, when the field scale length is large compared with a typical gyroradius. We demonstrate that the nonperturbative guiding center model makes exact predictions in these cases, even though it eliminates the cyclotron timescale, thereby establishing a theoretical baseline for performance of the nonperturbative formalism.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Finite orbit width effects in large aspect ratio stellarators

New orbit-averaged equations for low collisionality neoclassical fluxes in large aspect ratio stellarators with mirror ratios close to unity are derived. The equations retain finite orbit width effects by employing the second adiabatic invariant $J$ as a velocity-space coordinate and they have been implemented in the orbit-averaged neoclassical code KNOSOS (Velasco et al. , J. Comput. Phys. , vol. 418, 2020, 109512; Velasco et al. , Nucl. Fusion , vol. 61, 2021, 116013). The equations are used to study the $1/\nu$ regime and the lower collisionality regimes. For generic large aspect ratio stellarators with mirror ratios close to unity, as the collision frequency decreases, the $1/\nu$ regime transitions directly into the $\nu$ regime, without passing through a $\sqrt {\nu }$ regime. An explicit formula for the neoclassical fluxes in the $\nu$ regime is obtained. The formula includes the effect of particles that transition between different types of wells. While these transitions produce stochastic scattering independent of the value of the collision frequency in velocity space, the diffusion in real space remains proportional to the collision frequency. The $\sqrt {\nu }$ regime is only recovered in large aspect ratio stellarators close to omnigeneity: large aspect ratio stellarators with large mirror ratios and optimized large aspect ratio stellarators with mirror ratios close to unity. Neoclassical transport in large aspect ratio stellarators with large mirror ratios can be calculated with the orbit-averaged equations derived by Calvo et al. ( Plasma Phys. Control. Fusion , vol. 59, 2017, 055014). In these stellarators, the $\sqrt {\nu }$ regime exists in the collisionality interval $(a/R) \ln (R/a) \ll \nu _{ii} R a/\rho _i v_{ti} \ll R/a$ . In optimized large aspect ratio stellarators with mirror ratios close to unity, the $\sqrt {\nu }$ regime occurs in an interval of collisionality that depends on the deviation from omnigeneity $\delta$ : $\delta ^{2} |\ln \delta | \ll \nu _{ii} R a/\rho _i v_{ti} \ll 1$ . Here, $\nu _{ii}$ is the ion–ion collision frequency, $\rho _i$ and $v_{ti}$ are the ion gyroradius and thermal speed, and $a$ and $R$ are the minor and major radii.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Electron kinetics in a high- Z plasmoid

The problem of the electron dynamics on a closed magnetic field line passing through a high-Z plasmoid is considered. The electron kinetic equation is integrated over bounce motion and pitch angle, reducing the independent variables to a single adiabatic invariant plus time. Integration of the full Landau self-collision operator is carried out exactly, resulting in a nonlinear integro-differential operator in the new invariant. Conservation laws and the H theorem of the integrated self-collision operator are proven. Numerical solutions of the integrated kinetic equation are obtained with a self-consistent quasineutral electric potential, given the initial condition of a cold plasmoid immersed in a hot ambient plasma. The fact that cold electrons are deeply trapped in a potential with a parabolic peak leads to exactly 3/4 the usual rate of collisional heating by the ambient plasma, independent of any other parameters.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

Is there an exact magnetic moment for charged particle motion in a time-dependent, homogeneous magnetic field?

The non-perturbative guiding-centre model provides an exact alternative to full-orbit simulations of charged particle dynamics in situations where traditional guiding-centre theory may fail. We demonstrate that the charged particle motion in a homogeneous, time-varying magnetic field is a solvable example of the non-perturbative guiding-centre model. This entails showing that the exact magnetic moment of Qin and Davidson can be constructed to be asymptotic to the adiabatic invariant series of Kruskal. In contrast to the perturbative invariant, the exact invariant contains information about parametric resonances. These resonances destroy the conservation of the usual magnetic moment over very long times. This refutes some previous claims about the all-time invariance of the magnetic moment.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Modeling the Effects of Drift Orbit Bifurcation on Radiation Belt Electrons

Abstract Drift orbit bifurcation (DOB) has been suggested to play a major role in the loss and transport of radiation belt electrons since it violates the second adiabatic invariants of particles and makes the third invariant undefined. Results from our guiding‐center test particle simulations using the Tsyganenko‐1989c magnetic field model show that the DOB could affect a broad region of the outer radiation belt, which can penetrate inside the geosynchronous orbit at Kp ≥ 3, and its effects are more significant further away from the Earth, at a higher Kp and for higher electron energies. Specifically, the short‐term simulation results after one electron drift show both traditional and nontraditional DOB transport of electrons, with the nontraditional DOB, caused by a third minimum of the magnetic field strength near the equator, reported here for the first time. Moreover, our results show large ballistic jumps in the second invariant and radial distance for electrons at high equatorial pitch angles after one drift. In addition, long‐term DOB transport coefficients of electrons over many drifts are calculated based on our simulation results. We find that the pitch angle and radial diffusion coefficients of electrons due to DOB could be comparable to or even larger than those caused by electron interactions with chorus and Ultra‐Low‐Frequency waves, respectively. In sum, our results demonstrate that DOB could cause effective loss and transport of radiation belt electrons even in the absence of waves.

Astronomy & Astrophysics↗
DOE OSTI2024

Simulating Long–Term Dynamics of Radiation Belt Electrons Using DREAM3D Model

Here, we compared the performance of DREAM3D simulations in reproducing the long-term radiation belt dynamics observed by Van Allen Probes over the entire year of 2017 with various boundary conditions (BCs) and model inputs. Specifically, we investigated the effects of three different outer boundary conditions, two different low-energy boundary conditions for seed electrons, four different radial diffusion (RD) coefficients (DLL), four hiss wave models, and two chorus wave models from the literature. Using the outer boundary condition driven by GOES data, our benchmark simulation generally well reproduces the observed radiation belt dynamics inside L* = 6, with a better model performance at lower μ than higher μ, where μ is the first adiabatic invariant. By varying the boundary conditions and inputs, we find that: (a) The data-driven outer boundary condition is critical to the model performance, while adding in the data-driven seed population doesn't further improve the performance. (b) The model shows comparable performance with DLL from Brautigam and Albert (2000, https://doi.org/10.1029/1999ja900344), Ozeke et al. (2014, https://doi.org/10.1002/2013ja019204), and Liu et al. (2016, https://doi.org/10.1002/2015gl067398), while with DLL from Ali et al. (2016, https://doi.org/10.1002/2016ja023002) the model shows less RD compared to data. (c) The model performance is similar with data-based hiss models, but the results show faster loss is still needed inside the plasmasphere. (d) The model performs similarly with the two different chorus models, but better capturing the electron enhancement at higher μ using the Wang et al. (2019, https://doi.org/10.1029/2018ja026183) model due to its stronger wave power, since local heating for higher energy electrons is under-reproduced in the current model.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Outer Radiation Belt Dynamics During the October 2012 Storm Revisited: Rapid Inward Radial Transport From a Dynamic Outer Boundary

Earth's outer radiation belt electron flux is highly variable and can be enhanced by over an order of magnitude over timescales less than one day, as observed during the October 2012 storm. Previous studies of this storm (e.g., Reeves et al., 2013, https://doi.org/10.1126/science.1237743) have invoked local acceleration to explain this. However, here, we argue that the observations can instead be explained by fast inward radial transport. One method often invoked to distinguish between these two acceleration processes is the existence of local peaks in electron phase space density (PSD) as a function of L* at fixed first, M, and second, K, adiabatic invariants. However, this method relies on the assumption that the evolution of the PSD as a function of L* occurs over timescales slower than the satellite orbital period. Here, high spatiotemporal resolution data from the Global Positioning System (GPS) spacecraft constellation is used to show that enhancements in the PSD occur during the October 2012 storm over short timescales not resolvable by the Van Allen Probes. In addition, Geostationary Operational Environmental Satellite spacecraft data also indicate that these enhancements are consistent with relativistic electron injections. A radial diffusion model is shown to reproduce the PSD dynamics observed by the Van Allen Probes, once rapid variations at the simulation outer boundary are included, consistent with GPS data. This verifies that apparently “locally growing” peaks in PSD along high apogee satellite orbits can be produced by fast inward radial transport without requiring the action of any local acceleration processes.

99 GENERAL AND MISCELLANEOUS↗