Pitch Angle Diffusion Coefficients Research Articles

Pitch-Angle Diffusion of Radiation Belt Electrons and Precipitating Particle Fluxes: Dependence on VLF Wavefield Parameters

The dependence of the pitch-angle diffusion efficiency of energetic electrons in the Earth’s magnetosphere on the distribution of the whistler wave field along the geomagnetic flux tube is quantitatively studied for parameters corresponding to the location of the Sura and HAARP HF heating facilities. The expansion of the precipitation energy range with the increase of the region of geomagnetic latitudes occupied by the waves is shown. Using the calculated pitch-angle diffusion coefficient for a given spectrum of waves and their distribution along the flux tube, the ratio of the fluxes of precipitating and trapped particles at low altitude is determined. It is shown that at typical wave intensities corresponding to chorus VLF waves and plasmaspheric hiss, the fluxes of precipitating and trapped electrons can be comparable to each other. At the same time, for the wave amplitudes observed as a result of the action of heating facilities, the flux of precipitating electrons is negligible.

Field Line Curvature (FLC) Scattering in the Dayside Off‐Equatorial Minima Regions

AbstractMagnetic field line curvature (FLC) scattering is an effective mechanism for collisionless particle scattering. In the terrestrial magnetosphere, the FLC scattering plays an essential role in shaping the outer boundary of protons radiation belt, the rapid decay of ring current, and the formation of proton isotropic boundary (IB). However, previous studies have yet to adequately investigate the influence of FLC scattering on charged particles in the Earth's dayside magnetosphere, particularly in the off‐equatorial magnetic minima regions. This study employs T89 magnetic field model to investigate the impacts of FLC scattering on ring current protons in the dayside magnetosphere, with a specific focus on the off‐equatorial minimum regions. We analyze the spatial distributions of single and dual magnetic minima regions, adiabatic parameter, and pitch angle diffusion coefficients due to FLC scattering as functions of Kp. The results show that the effects of FLC scattering are significant not only on the dusk and dawn sides but also in the off‐equatorial minima regions on the noon. Additionally, we investigate the role of dipole tilt angle in the hemispheric asymmetry of FLC scattering effects. The dipole tilt angle controls the overall displacement of the dayside magnetosphere, resulting in different FLC scattering effects in the two hemispheres. Our study holds significance for understanding the FLC scattering effects in the off‐equatorial region of Earth's dayside magnetosphere and for constructing a more accurate dynamic model of particles.

Particle Scattering in the Presence of Moderate to Strong Adiabatic Focusing

AbstractStarting from the Vlasov‐Boltzmann equation, we derive corrections to the quasi linear Fokker‐Planck pitch angle scattering coefficient caused by a non‐uniformity of the background magnetic field. Integration of the perturbed particle distribution function is performed along a modified unperturbed trajectory that takes into account the effects of focusing and mirroring on pitch angle and gyrophase. We calculate the changes to the quasi‐linear scattering rate owing to a variations in the field strength and the gyrophase drift that affects the resonance between particles and magnetic fluctuations. It is shown that the scattering coefficient has an oscillating behavior in pitch angle when the focusing lengthscale is comparable to the correlation length of the turbulence and the particle's Larmor radius. The applicability of analytic results are demonstrated with a simple model of charged particles trapped in a magnetic field of a planet. It is suggested that the pitch angle diffusion coefficient could become negative for the case of very strong focusing.

Statistical Properties of Exohiss Waves and Associated Scattering Losses of Radiation Belt Electrons

AbstractExohiss waves are a type of structureless whistler‐mode waves that exist in the low density plasmatrough outside the plasmapause and may potentially perturb the motions of electrons in the radiation belt and ring current. Using data from Van Allen Probe A, we analyze the distribution of magnetic power spectral density (PSD) of exohiss waves in different magnetic local time (MLT) and L‐shell regions near the geomagnetic equator. The results reveal that the peak magnetic PSD of exohiss waves is the weakest at MLT = 0–6 and the strongest at MLT = 12–18. The magnetic PSDs of exohiss waves are much lower than those of chorus and hiss waves except for the MLT = 12–18 sector. In addition, we calculated the quasi‐linear bounce‐averaged pitch angle and momentum diffusion coefficients (〈Dαα〉 and 〈Dpp〉) of electrons caused by exohiss waves. The diffusion coefficients are then compared with those caused by chorus and hiss waves. The peak 〈Dαα〉 of electrons driven by exohiss waves becomes stronger as L‐shell increases at all MLTs and is the greatest on the dayside, especially in the sector of MLT = 12–18. Exohiss waves have more significant effect on the loss of radiation belt electrons with specific energy levels related to MLT and L‐shell region compared to chorus and hiss waves. On the other hand, 〈Dpp〉 of electrons caused by exohiss waves is very small, which illustrates that exohiss waves have almost no acceleration effect on electrons.

Characterizing Radiation‐Belt Energetic Electron Precipitation Spectra: A Comparison of Quasi‐Linear Diffusion Theory With In Situ Measurements

AbstractHigh energy electron precipitation from the Earth's radiation belts is important for loss from the radiation belts and atmospheric chemistry. We follow up investigations presented in Reidy et al. (2021, https://doi.org/10.1029/2020ja028410) where precipitating flux is calculated inside the field of view of the POES T0 detector using quasi‐linear theory and pitch angle diffusion coefficients (Dαα) from the British Antarctic Survey (BAS). These results showed good agreements at >30 keV for L* >5 on the dawnside but the flux were too low at higher energies. We have investigated the effect of changing parameters in the calculation of the precipitating flux to improve the results for the higher energies using comparisons of in situ flux and cold plasma measurements from GOES‐15 and RBSP. We find that the strength of the diffusion coefficients rather than the shape of the source spectrum has the biggest effect on the calculated precipitation. In particular we find decreasing the cold plasma density used in the calculation of Dαα increases the diffusion and hence the precipitation at the loss cone for the higher energies, improving our results. The method of calculating Dαα is also examined, comparing co‐located rather than averaged RBSP measurements. We find that the method itself has minimal effect but using RBSP derived Dαα improved our results over using Dαα calculated using the entire BAS wave data base; this is potentially due to better measurements of the cold plasma density from RBSP than the other spacecraft included in the BAS wave data base (e.g., THEMIS).

Parametrization of coefficients for sub-grid modeling of pitch-angle diffusion in global magnetospheric hybrid-Vlasov simulations

Sub-grid models are key tools to accurately describe the physical processes at play in a system when high-resolution simulations are not feasible. We previously developed a sub-grid model for pitch-angle diffusion in hybrid-Vlasov simulations of Earth's magnetosphere. However, a more precise description of the pitch-angle diffusion coefficient is required to apply this model to global simulations. In this study, we use an existing method to parametrize pitch-angle diffusion coefficients from monotonic distribution functions and adapt it to bi-Maxwellian distributions. We determine these coefficients for various values of the ion temperature anisotropy and plasma β∥. We use these newly parametrized coefficients in our sub-grid model and show that it accurately models reduction of temperature anisotropy in both local simulations and global simulations of the Earth's magnetosphere, while using minimal computational resources.

Simulation of radiation belt wave-particle interactions in an MHD-particle framework

In this paper we describe K2, a comprehensive simulation model of Earth’s radiation belts that includes a wide range of relevant physical processes. Global MHD simulations are combined with guiding-center test-particle methods to model interactions with ultra low-frequency (ULF) waves, substorm injections, convective transport, drift-shell splitting, drift-orbit bifurcations, and magnetopause shadowing, all in self-consistent MHD fields. Simulation of local acceleration and pitch-angle scattering due to cyclotron-scale interactions is incorporated by including stochastic differential equation (SDE) methods in the MHD-particle framework. The SDEs are driven by event-specific bounce-averaged energy and pitch-angle diffusion coefficients. We present simulations of electron phase-space densities during a simplified particle acceleration event based on the 17 March 2013 event observed by the Van Allen Probes, with a focus on demonstrating the capabilities of the K2 model. The relative wave-particle effects of global scale ULF waves and very-low frequency (VLF) whistler-mode chorus waves are compared, and we show that the primary acceleration appears to be from the latter. We also show that the enhancement with both ULF and VLF processes included exceeds that of VLF waves alone, indicating a synergistic combination of energization and transport processes may be important.

Efficient Scattering Loss of Energetic Electrons by Enhanced Higher‐Band ECH Waves Observed by Van Allen Probes

AbstractElectron cyclotron harmonic (ECH) waves, which are responsible for the diffuse aurora, are usually observed with the strongest intensity in the first band. Here, we present observations of enhanced higher‐band ECH with wave intensities above 10−3 mV2m−2 Hz−1 by Van Allen Probes. Fully thermal simulation results indicate that these waves can be locally excited by the observed electron distributions, and higher plasma density and lower magnetic strength are favorable for generating the enhanced higher‐band ECH. The pitch angle diffusion coefficient of the higher‐band ECH can exceed 10−5 s−1 inside the loss cone for energetic electrons (300 eV–∼15 keV), greater than the momentum diffusion coefficient by ∼100 times. It is suggested that the enhanced higher‐band ECH can also efficiently scatter these electrons into the loss cone and contribute to the diffuse aurora.

The Effect of Compression Induced Chorus Waves on 10–100 s eV Electron Precipitation

AbstractOn 7 January 2014, a solar storm erupted, which eventually compressed the Earth's magnetosphere leading to the generation of chorus waves. These waves enhanced local wave‐particle interactions and led to the precipitation of electrons from 10 s eV to 100 s keV. This paper shows observations of a low energy cutoff in the precipitation spectrum from Van Allen Probe B Helium Oxygen Proton Electron measurements. This low energy cutoff is well replicated by the predicted loss calculated from pitch angle diffusion coefficients from wave and plasma observations on Probe B. To our knowledge, this is the first time a single spacecraft has been used to demonstrate an accurate theoretical prediction for chorus wave‐induced precipitation and its low energy cutoff. The specific properties of the precipitating soft electron spectrum have implications for ionospheric activity, with the lowest energies mainly contributing to thermospheric and ionospheric upwelling, which influences satellite drag and ionospheric outflow.

Modeling the Effects of Drift Orbit Bifurcation on Radiation Belt Electrons

AbstractDrift 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.

Inner Belt Wisp Precipitation Measured by ELFIN: Regimes of Energetic Electron Scattering by VLF Transmitter Waves

AbstractMan‐made very low frequency (VLF) transmitter waves play a critical role in energetic electron scattering and precipitation from the inner radiation belt, a type of which is called wisp precipitation. Wisps exhibit dispersive energy‐versus‐L spectra due to the evolution of electron cyclotron resonance conditions with near‐monochromatic VLF transmitter waves. Here, we report on such observations of inner belt wisp precipitation events with full pitch angle resolution in the energy range of 50 to ∼500 keV as measured by Electron Loss and Fields Investigation (ELFIN) at L < ∼2 between March 2021 and April 2022. Statistical observations (82 events) reveal occasional (18 events) wisp precipitation events with local bounce‐loss‐cone electron flux enhancements, which provide new information compared with flux enhancements measured in previous studies only in the drift loss cone. Based on magnetic field and plasmaspheric density models, quasilinear theory, and detailed pitch angle distributions of wisps from ELFIN, we have estimated the wisp electron bounce‐averaged pitch angle diffusion coefficients to be of the order of 10−4 to 10−2 s−1. These are several orders of magnitude larger than the diffusion rates calculated from models using global statistical averages of VLF transmitter wave power. When using our estimated diffusion coefficients to deduce the associated local transmitter wave amplitudes near the equator, based on quasilinear calculations from a transmitter‐induced electron diffusion model, we find these wave amplitudes to be >1 mV/m. Although probable overestimates, such inferred wave amplitudes exceed the theoretical threshold amplitude for nonlinear interactions, strongly suggesting that it is necessary to include nonlinear effects for an accurate evaluation of energetic electron scattering by transmitter waves.

FARWEST: Efficient Computation of Wave‐Particle Interactions for a Dynamic Description of the Electron Radiation Belt Diffusion

AbstractIn this paper, we present a new method to compute rapidly, wave particle interaction induced pitch angle and momentum diffusion coefficients. Those terms are normally obtained by the numerical solving of equations based on the quasi‐linear theory. However, this bulk resolution leads to a high computational cost, preventing the integration of plasma density and VLF waves nowcasts and forecasts. Therefore, and in the context of the SAFESPACE project (H2020), we implemented a new wave particle interaction code called FARWEST (FAst Radiation diffusion with Waves ESTimator), with an efficient interpolation based method. The proposed implementation was validated against reference results obtained by WAPI, ONERA's wave particle interaction legacy code, with a substantial reduction of the computing time while conserving the physical accuracy of the generated coefficients. The FARWEST code is later used to measure the impact of a time‐dependent plasma density distribution (numerically computed by BIRA's plasmaspheric model SPM) on the coefficients and on the representation of the electron flux map by Salammbô, ONERA's radiation belt code.

Quantifying Radiation Belt Electron Loss Processes at L

We present a comprehensive analysis of the processes that lead to quasilinear pitch-angle-scattering loss of electrons from the L<4 region of the Earth's inner magnetosphere during geomagnetically quiet times. We consider scattering via Coulomb collisions, hiss waves, lightning-generated whistler (LGW) waves, waves from ground-based very-low-frequency (VLF) transmitters, and electromagnetic ion cyclotron (EMIC) waves. The amplitude, frequency, and wave normal angle spectra of these waves are parameterized with empirical wave models, which are then used to compute pitch-angle diffusion coefficients. From these coefficients, we estimate the decay timescales, or lifetimes, of 30keV to 4MeV electrons and compare the results with timescales obtained from in-situ observations. We demonstrate good quantitative agreement between the two over most of the L and energy range under investigation. Our analysis suggests that the electron decay timescales are very sensitive to the choice of plasmaspheric density model. At L<2, where our theoretical lifetimes do not agree well with the observations, we show that including Coulomb energy drag (ionization energy loss) in our calculations significantly improves the quantitative agreement with the observed decay timescales. We also use an accurate model of the geomagnetic field to provide an estimate of the effect that the drift-loss cone has on the theoretically calculated electron lifetimes, which are usually obtained using an axisymmetric dipole field.

Energetic electron scattering by kinetic Alfvén waves at strong magnetic field gradients of dipolarization front

Energetic electron scattering and precipitation from the Earth's plasma sheet to the ionosphere is an important contributor to magnetosphere–ionosphere coupling. In this study, we investigate the role of one of the most intense wave emissions, kinetic Alfvén waves (KAWs), in energetic electron scattering. We have evaluated the effect of KAWs on energetic electrons within a curved magnetic field configuration exhibiting sharp cross field gradients. The magnetic field in Earth's magnetotail plasma sheet with an embedded dipolarization front is used as a working example. Taking into account electron bounce motion and perpendicular guiding-center drifts, we have shown that electrons with energies of tens to hundreds of keV can be scattered by KAWs in pitch angle and momentum through Doppler-shifted Landau resonance near the magnetic equator. The bounce-averaged pitch-angle diffusion coefficients for near-loss-cone (∼2°) electrons are on the order of 10–7–10–6 rad2/s for a characteristic KAW amplitude of 1 mV/m and approach the strong diffusion limit of ∼10–4 rad2/s for amplitudes of greater than 10 mV/m. These results suggest that under such ambient conditions, KAWs can pitch-angle scatter energetic electron population into the loss cone. In Earth's plasma sheet, this scattering is, thus, very likely to cause significant precipitation during active times. The diffusion coefficients of energetic electrons at large pitch angles (∼45°–∼80°) are more than two orders of magnitude larger than those of electrons near the loss cone, suggesting that KAWs contribute to isotropization of anisotropic electrons due to adiabatic heating should they drift into the vicinity of the magnetic field gradient.

Electron Scattering by Very‐Low‐Frequency and Low‐Frequency Waves From Ground Transmitters in the Earth's Inner Radiation Belt and Slot Region

AbstractThe very‐low frequency (VLF) and low frequency (LF) waves from ground transmitters propagate in the ionospheric waveguide, and a portion of their power leaks to the Earth's inner radiation belt and slot region where it can cause electron precipitation loss. Using Van Allen Probes observations, we perform a survey of the VLF and LF transmitter waves at frequencies from 14 to 200 kHz. The statistical electric and magnetic wave amplitudes and frequency spectra are obtained at 1 &lt; L &lt; 3. Based on a recent study on the propagation of VLF transmitter waves, we divide the total wave power into ducted and unducted portions, and model the wave normal angle of unducted waves with dependences on L shell, magnetic latitude, and wave frequency. At lower frequencies, the unducted waves are launched along the vertical direction and the wave normal angle increases during the propagation until reaching the Gendrin angle; at higher frequencies, the normal angle of unducted waves follows the variation of Gendrin angle. We calculate the bounce‐averaged pitch angle and momentum diffusion coefficients of electrons due to ducted and unducted VLF and LF waves. Unducted and ducted waves cause efficient pitch angle scattering at L = 1.5 and 2.5, respectively. Although the wave power from ground transmitters at frequencies higher than 30 kHz is low, these waves can cause the pitch angle scattering of lower energy (2–200 keV at L = 1.5) electrons, which cannot resonate with the VLF transmitter waves at frequencies below 30 kHz, lightning generated whistlers, or plasmaspheric hiss.

A Comparison with Theory of Computation and Estimation of Pitch-Angle Diffusion Coefficients from Simulations in Noisy Reduced Magnetohydrodynamic Turbulence

The transport of energetic charged particles in turbulent magnetic fields is a topic of interest in various astrophysical and laboratory plasma contexts. In order to estimate the mean free path λ∥ of a particle in the direction parallel to the mean magnetic field, one can use theoretical expressions that include the pitch-angle diffusion coefficient Dμμ. In this work we evaluate theories for Dμμ in the context of the noisy reduced magnetohydrodynamic (NRMHD) model where turbulent fluctuations are absent at large parallel wavenumbers. For most turbulence models, the standard quasilinear theory predicts zero pitch-angle diffusion only for particles with a 90◦ pitch angle, but for NRMHD a range of pitch angles is affected, leading to infinite λ∥. We examine two theories that include resonance broadening which yield finite λ∥ and compare them with test-particle computer simulations in which the parallel mean free path can be readily obtained. We find that both theories are quite accurate in some regions of the parameter space considered, but neither is particularly good when the particle Larmor radius RL becomes much smaller than the quasilinear resonance limit.

Machine Learning Methods Applied to the Global Modeling of Event-Driven Pitch Angle Diffusion Coefficients During High Speed Streams

Whistler-mode waves in the inner magnetosphere cause electron precipitation in the atmosphere through the physical process of pitch-angle diffusion. The computation of pitch-angle diffusion relies on quasi-linear theory and becomes time-consuming as soon as it is performed at high temporal resolution from satellite measurements of ambient wave and plasma properties. Such an effort is nevertheless required to capture accurately the variability and complexity of atmospheric electron precipitation, which are involved in various Earth’s ionosphere-magnetosphere coupled problems. In this work, we build a global machine-learning model of event-driven pitch-angle diffusion coefficients for storm conditions based on the data of a variety of storms observed by the NASA Van Allen Probes. We first proceed step-by-step by testing 8 nonparametric machine learning methods. With them, we derive machine learning based models of event-driven diffusion coefficients for the storm of March 2013 associated with high-speed streams. We define 3 diagnostics that allow highlighting of the properties of the selected model and selection of the best method. Three methods are retained for their accuracy/efficiency: spline interpolation, the radial basis method, and neural networks (DNN), the latter being selected for the second step of the study. We then use event-driven diffusion coefficients computed from 32 high-speed stream storms in order to build for the first time a statistical event-driven diffusion coefficient that is embedded within the retained DNN model. We achieve a global mean event-driven model in which we introduce a two-parameter dependence, with both the Kp-index and time kept as in an epoch analysis following the storm evolution. The DNN model does not entail any issue to reproduce quite perfectly its target, i.e., averaged diffusion coefficients, with rare exception in the Landau resonance region. The DNN mean model is then used to analyze how mean diffusion coefficients behave compared with individual ones. We find a poor performance of any mean models compared with individual events, with mean diffusion coefficients computing the general trend at best, due to their large variability. The DNN-based model allows simple and fast data exploration of pitch-angle diffusion among its multiple variables. We finally discuss how to conduct uncertainty quantification of Fokker-Planck simulations of storm conditions for space weather nowcasting and forecasting.

Numerical study of cosmic ray confinement through dust resonant drag instabilities

ABSTRACT We investigate the possibility of cosmic ray (CR) confinement by charged dust grains through resonant drag instabilities (RDIs). We perform magnetohydrodynamic particle-in-cell simulations of magnetized gas mixed with charged dust and cosmic rays, with the gyro-radii of dust and GeV CRs on ∼au scales fully resolved. As a first study, we focus on one type of RDI wherein charged grains drift super-Alfvénically, with Lorentz forces strongly dominating over drag forces. Dust grains are unstable to the RDIs and form concentrated columns and sheets, whose scale grows until saturating at the simulation box size. Initially perfectly streaming CRs are strongly scattered by RDI-excited Alfvén waves, with the growth rate of the CR perpendicular velocity components equaling the growth rate of magnetic field perturbations. These rates are well-predicted by analytic linear theory. CRs finally become isotropized and drift at least at ∼vA by unidirectional Alfvén waves excited by the RDIs, with a uniform distribution of the pitch angle cosine μ and a flat profile of the CR pitch angle diffusion coefficient Dμμ around μ = 0, without the ‘90○ pitch angle problem.’ With CR feedback on the gas included, Dμμ decreases by a factor of a few, indicating a lower CR scattering rate, because the backreaction on the RDI from the CR pressure adds extra wave damping, leading to lower quasi-steady-state scattering rates. Our study demonstrates that the dust-induced CR confinement can be very important under certain conditions, e.g. the dusty circumgalactic medium around quasars or superluminous galaxies.

Effects of Hot Protons on the Pitch Angle Scattering of Ring Current Protons by EMIC Waves

AbstractVia cyclotron resonant interactions, electromagnetic ion cyclotron (EMIC) waves play an important role in the loss of ring current protons. In this study, by calculating the proton bounce‐averaged pitch angle diffusion coefficients using both the cold and hot plasma dispersion relations, we investigate the effects of hot protons on the EMIC wave‐induced scattering loss of ring current protons. Our results show that, for H+ band (He+ band) EMIC waves, inclusion of hot protons results in significant decrease of pitch angle diffusion coefficients of ∼10–60 keV (4–30 keV) protons, while the scattering efficiency of higher energy protons increases at low pitch angles and decreases at relatively high pitch angles. We also find that the cold plasma approximation seriously underestimates the loss timescales of protons at energies from a few keV to tens of keV but overestimate that of higher energy protons. The differences in proton loss timescales caused by hot protons are generally less than a factor of ∼5 for H+ band but can exceed an order of magnitude for He+ band, showing a strong dependence on the hot proton concentration and temperature anisotropy and L‐shell. This study confirms that effects of hot protons play a crucial role in the EMIC wave driven loss of ring current protons and should be included in future modeling of ring current dynamics.

Determining Pitch-Angle Diffusion Coefficients for Electrons in Whistler Turbulence

Transport of energetic electrons in the heliosphere is governed by resonant interaction with plasma waves, for electrons with sub-GeV kinetic energies specifically with dispersive modes in the whistler regime. In this paper, particle-in-cell simulations of kinetic turbulence with test-particle electrons are performed. The pitch-angle diffusion coefficients of these test particles are analyzed and compared to an analytical model for left-handed and right-handed polarized wave modes.