Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Magnetic field modeling”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records

Measurements of the time-dependent cosmic-ray Sun shadow with seven years of IceCube data: Comparison with the Solar cycle and magnetic field models

Observations of the time-dependent cosmic-ray Sun shadow have been proven as a valuable diagnostic for the assessment of solar magnetic field models. Here, seven years of IceCube data are compared to solar activity and solar magnetic field models. A quantitative comparison of solar magnetic field models with IceCube data on the event rate level is performed for the first time. Additionally, a first energy-dependent analysis is presented and compared to recent predictions. We use seven years of IceCube data for the moon and the Sun and compare them to simulations on data rate level. The simulations are performed for the geometrical shadow hypothesis for the moon and the Sun and for a cosmic-ray propagation model governed by the solar magnetic field for the case of the Sun. We find that a linearly decreasing relationship between Sun shadow strength and solar activity is preferred over a constant relationship at the 6.4σ level. We test two commonly used models of the coronal magnetic field, both combined with a Parker spiral, by modeling cosmic-ray propagation in the solar magnetic field. Both models predict a weakening of the shadow in times of high solar activity as it is also visible in the data. We find tensions with the data on the order of 3σ for both models, assuming only statistical uncertainties. The magnetic field model CSSS fits the data slightly better than the PFSS model. This is generally consistent with what is found previously by the Tibet AS-γ Experiment; a deviation of the data from the two models is, however, not significant at this point. Regarding the energy dependence of the Sun shadow, we find indications that the shadowing effect increases with energy during times of high solar activity, in agreement with theoretical predictions.

79 ASTRONOMY AND ASTROPHYSICS↗

Modeling magnetic fields with helical solutions to Laplace’s equation

The series solution to Laplace's equation in a helical coordinate system is derived and refined using symmetry and chirality arguments. These functions and their more commonplace counterparts are used to model solenoidal magnetic fields via linear, multidimensional curve-fitting. A judicious choice of functional forms, a small number of free parameters and sparse input data can lead to highly accurate, fine-grained modeling of solenoidal magnetic fields, including helical features arising from the winding of the solenoid, with overall field accuracy at better than one part per million.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

An Integrated Assessment of a G3 GMD Event on Large-Scale Power Grids: From Magnetometer Data to Geomagnetically Induced Current Analysis

Solar activities can cause geomagnetic disturbances (GMDs) that give rise to geomagnetically induced currents (GICs) which may compromise the reliability of the power system. In order to build more reliable models representing GMD interactions with the power grid, the power system’s detailed electrical model must be considered along with fluctuations in the earth’s magnetic and induced surface electric fields. Here, this study investigates the impact of incorporating spatially varying magnetic fields into surface electric field models on GMD risk metrics. A spatially independent magnetic field model and a spatially varying model are compared through simulations. To perform this analysis, the earth’s magnetic field disturbances are transformed into surface electric fields using respective one-dimensional earth conductivity models. Then, the modeling impact of these electric fields is studied using a 2,000-bus grid for Texas and a 25,000-bus grid for the northeast and mid- Atlantic regions of the United States. Simulation results reveal that the inclusion of spatially varying magnetic fields results in considerable differences in GMD risk metrics, highlighting the importance of accounting for spatial variability when assessing GMD risks in the power system.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Pro‐ L * ‐ A Probabilistic L * Mapping Tool for Ground Observations

Abstract Both ground and space observations are used extensively in the modeling of space weather processes within the Earth’s magnetosphere. In radiation belt physics modeling, one of the key phase‐space coordinates is L *, which indicates the location of the drift paths of energetic electrons. Global magnetic field models allow a subset of locations on the ground (mainly subauroral) to be mapped along field lines to a location in space and transformed into L *, provided that the initial ground location maps to a closed drift path. This allows observations from ground, or low‐altitude space‐based platforms to be mapped into space in order to inform radiation belt modeling. Many data‐based magnetic field models exist; however, these models can significantly disagree on mapped L * values for a single point on the ground, during both quiet times and storms. We present a state of the art probabilistic L * mapping tool, Pro‐ L *, which produces probability distributions for L * corresponding to a given ground location. Pro‐ L * has been calculated for a high resolution magnetic latitude by magnetic local time grid in the Earth’s Northern Hemisphere. We have developed the probabilistic model using 11 years of L * calculations for seven widely used magnetic field models. Usage of the tool is highlighted for both event studies and statistical models, and we demonstrate a number of potential applications.

79 ASTRONOMY AND ASTROPHYSICS↗

Modelling pulsed field magnetization of iron-based bulk superconductors

Abstract Bulk superconductors can be used as super-strength quasi-permanent magnets capable of providing magnetic flux densities considerably superior to conventional permanent magnets. This makes them attractive for several engineering applications that rely on strong magnetic fields like rotating machines, NMR/MRI and magnetic drug delivery systems. Recently, the authors reported a record trapped magnetic field in an iron-based bulk superconductor: 2.83 T was trapped in potassium-doped barium iron arsenide (Ba, K)Fe 2 As 2 (or Ba122) at 5 K. Of particular significance is that the strength and temporal stability of this magnetic field exceeds the requirements of MRI machines, indicating iron-based bulks can now perform at levels demanded by engineering applications. One crucial challenge for their practical use, however, is the need to apply and remove an external magnetic field to magnetize them. Pulsed field magnetization (PFM) shows great promise as a practical method of magnetizing bulks, but the process generates heat in the bulk that is detrimental to its superconducting performance and ability to act as a super-strength magnet. In this paper, coupled electromagnetic–thermal numerical models are used to simulate the PFM of iron-based bulk superconductors. Here we focus on the recent-record-breaking, fine-grain polycrystalline K-doped Ba122 bulks. The impact that the specific J c ( B ) characteristics and thermal properties of the Ba122 material—all of which have been experimentally measured from state-of-the-art samples—have on the magnetic flux dynamics and thermal behaviour during PFM, including the final trapped field, is investigated. We show that because the thermal properties are similar to those of REBa 2 Cu 3 O 7 −δ bulks, a similar response to pulsed fields is obtained. A maximum trapped field of ∼0.81 T (∼43.3% of the maximum trapped field capability under ideal, field-cooling conditions) was simulated at 5 K, with a magnetization efficiency of ∼54%. The modelling framework provides a fast and flexible tool for optimising the practical PFM process at different operating temperatures to maximise the trapped field in state-of-the-art Ba122 bulks and to guide the design of future experiments.

bulk superconductors↗

Modeling hydrodynamics, magnetic fields, and synthetic radiographs for high-energy-density plasma flows in shock-shear targets

Three-dimensional FLASH radiation-magnetohydrodynamics (radiation-MHD) modeling is carried out to study the hydrodynamics and magnetic fields in the shock-shear derived platform. Here, simulations indicate that fields of tens of Tesla can be generated via the Biermann battery effect due to vortices and mix in the counterpropagating shock-induced shear layer. Synthetic proton radiography simulations using MPRAD and synthetic X-ray image simulations using SPECT3D are carried out to predict the observable features in the diagnostics. Quantifying the effects of magnetic fields in inertial confinement fusion and high-energy-density plasmas represents frontier research that has far-reaching implications in basic and applied sciences.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Searches for New Physics With Muon Conversion at Fermilab and Triboson Production at the LHC

We report on several efforts to search for physics beyond the standard model of particle physics at broad energy scales. The Mu2e experiment at Fermilab will search for charged lepton flavor violation via the muon to electron conversion process, which is suppressed in the Standard Model. Mu2e will be operated at a low energy, yet can probe New Physics at very high mass scales (O(1e3 - 1e4 ) TeV). At high energies, the CMS experiment at the CERN LHC continues to deliver an impressive suite of Standard Model measurements and limits on a variety of New Physics signatures. Mu2e is under construction and slated to collect its first physics data in the coming years. This thesis describes work done during the construction phase of Mu2e and focuses on two critical areas: magnetic field modeling and statistical analysis. We describe a novel method for field modeling which we validate using a simulated dataset representing the expected magnetic field in the Detector Solenoid. This method blends a standard least-squares fitting technique that utilizes physically motivated analytical model functions with a novel physics informed network that is constructed to obey Maxwell’s equations. We show the technique can model the field with an accuracy of 10−7 despite the presence of injected noise in the pseudo-measurements at the 10−5 level. We then present preliminary results of the calibration of 3D Hall probes at the sub-10−4 level. These probes will be used to directly measure the Mu2e Detector Solenoid magnetic field on a sparse grid; these measurements serve as the input to the field model fitting. Finally, we describe the first implementation of both an unbinned shape analysis and a Bayesian interpretation applied to Mu2e pseudo-data. Up to 20% tighter limits can be set by the shape analysis compared to a standard cut & count analysis. The AlCap experiment collected data at PSI in 2015 to measure several important quantities related to nuclear muon capture on an aluminum target, which is a significant background process for Mu2e. The neutron emission from muon capture can introduce background hits in the Mu2e detectors and can increase radiation damage in various elements of the apparatus. We present measurements of the neutron group fluence and mean neutron multiplicity for muon capture on aluminum nuclei. Finally, we discuss an analysis of triboson production at CMS using an Effective Field Theory framework. Standard Model triboson production, which was first observed at CMS in 2020, has a relatively small cross section and provides direct access to both anomalous triple gauge couplings and quartic gauge couplings. These couplings, interpreted in the Standard Model Effective Field Theory, are studied in the present work. We target the boosted regime where the background rate is low and yields are enhanced when dimension-6 and dimension-8 Wilson coefficients are non-zero. We do not observe an excess in the data and therefore set bounds on the Wilson coefficients. For dimension-6 coefficients the tightest observed (expected) bounds are set on cW /Λ2 where Λ is the mass scale of new physics; the bounds are [−0.13, 0.12] TeV−2 ([−0.12, 0.12] TeV−2 ) at 95% CL. The tightest bounds in dimension-8 are set on fT,0 / Λ4 ; the observed (expected) bounds at 95% CL are [−0.63, 0.69] TeV−4 ([−0.54, 0.62] TeV−4 ). Additional results are presented which include scenarios where multiple Wilson coefficients are non-zero, the application of signal model clipping to address unitarity violation in Effective Field Theories, and a novel template fit developed for easier reinterpretation of our results.

Kampa, Cole Erik [Northwestern U. (main)] (ORCID:0↗

Effect of magneto-mechanical synergism in the process-structure correlation in Fe–C alloys: A phase-field modeling approach

Applied magnetic fields can alter phase equilibria and kinetics in steels; however, quantitatively resolving how magnetic, chemical, and elastic driving forces jointly influence the microstructure remains challenging. We develop a quantitative magneto-mechanically coupled phase-field model for the Fe–C system that couples a CALPHAD-based chemical free energy with demagnetization-field magnetostatics and microelasticity. Here, the model reproduces single- and multi-particle evolution during the α → γ inverse transformation at 1023 K under external fields up to 20 T, including ellipsoidal morphologies observed experimentally at 8 T. Chemically driven growth is isotropic; a magnetic interaction introduces an anisotropic driving force that elongates γ precipitates along the field into ellipsoids, while elastic coherency promotes faceting, yielding elongated cuboidal or “brick-like” particles under combined magneto-elastic coupling. Growth kinetics increase with C content, and decrease with field strength and misfit strain. Multi-particle simulations reveal dipolar interaction-mediated coalescence for field-parallel neighbors and ripening for field-perpendicular neighbors. Incorporating field-dependent diffusivity from experiment slows kinetics as expected; a first-principles-motivated anisotropic diffusivity correction is estimated to be small (<2%). These results establish a process-structure link for magnetically assisted heat treatments of Fe–C alloys and provide guidance for microstructure control via chemo-magneto-mechanical synergism.

Magnetic field↗

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↗

Structure of Self-Generated Magnetic Fields in Laser-Solid Interaction from Proton Tomography

Self-generated magnetic fields in laser-solid interactions are experimentally characterized to reveal the 3D location and local field strength, rather than path-integrated quantities, using multi-view proton radiography and tomographic inversion. We infer magnetic fields that extend several millimeters off the target into the hot, rarefied corona, sufficient to strongly magnetize the plasma (Ω e τ e ≫ 1). The data are compared to MHD simulations incorporating recent improvements in modeling magnetic field generation and transport; the volume-averaged coronal field strength and magnetic flux agree to within 25% using a model with magnetic re-localization of transport, although the near-target morphology is not reproduced. This work demonstrates tomographic proton radiography as a valuable tool for investigating magnetic fields in laser-produced plasmas.

High-energy-density plasmas↗

Large gyro-orbit model of ion velocity distribution in plasma near a wall in a grazing-angle magnetic field

A model is presented for the ion distribution function in a plasma at a solid target with a magnetic field $\boldsymbol {B}$ inclined at a small angle, $\alpha \ll 1$ (in radians), to the target. Adiabatic electrons are assumed, requiring $\alpha \gg \sqrt {Zm_{e}/m_{i}}$ , where $m_{e}$ and $m_{i}$ are the electron and ion mass, respectively, and $Z$ is the charge state of the ion. An electric field $\boldsymbol {E}$ is present to repel electrons, and so the characteristic size of the electrostatic potential $\phi$ is set by the electron temperature $T_{e}$ , $e\phi \sim T_{e}$ , where $e$ is the proton charge. An asymptotic scale separation between the Debye length $\lambda _{D} = \sqrt {\epsilon _0 T_{{e}} / e^{2} n_{{e}} }$ , the ion sound gyro-radius $\rho _{s} = \sqrt { m_{i} ( ZT_{e} + T_{i} ) } / (ZeB)$ and the size of the collisional region $d_{c} = \alpha \lambda _{\textrm {mfp}}$ is assumed, $\lambda _{D} \ll \rho _{s} \ll d_{c}$ . Here $\epsilon _0$ is the permittivity of free space, $n_{e}$ is the electron density, $T_{i}$ is the ion temperature, $B= |\boldsymbol {B}|$ and $\lambda _{\textrm {mfp}}$ is the collisional mean free path of an ion. The form of the ion distribution function is assumed at distances $x$ from the wall such that $\rho _{s} \ll x \ll d_{c}$ , that is, collisions are not treated. A self-consistent solution of the electrostatic potential for $x \sim \rho _{s}$ is required to solve for the quasi-periodic ion trajectories and for the ion distribution function at the target. The large gyro-orbit model presented here allows to bypass the numerical solution of $\phi (x)$ and results in an analytical expression for the ion distribution function at the target. It assumes that $\tau =T_{i}/(ZT_{e})\gg 1$ , and ignores the electric force on the quasi-periodic ion trajectory until close to the target. For $\tau \gtrsim 1$ , the model provides an extremely fast approximation to energy–angle distributions of ions at the target. These can be used to make sputtering predictions.

Physics↗

An adjoint method for determining the sensitivity of island size to magnetic field variations

An adjoint method to calculate the gradient of island width in stellarators is presented and applied to a set of magnetic field configurations. The underlying method for calculation of the island width is that of Cary & Hanson ( Phys. Fluids B, vol. 3, issue 4, 1991, pp. 1006–1014) (with a minor modification), and requires that the residue of the island centre be small. Therefore, the gradient of the residue is calculated in addition. Both the island width and the gradient calculations are verified using an analytical magnetic field configuration introduced by Reiman & Greenside ( Comput. Phys. Commun. , vol. 43, issue 1, 1986, pp. 157–167). The method is also applied to the calculation of the shape gradient of the width of a magnetic island in a National Compact Stellarator Experiment (NCSX) vacuum configuration with respect to positions on a coil. A gradient-based optimization is applied to a magnetic field configuration studied by Hanson & Cary ( Phys. Fluids , vol. 27, issue 4, 1984, pp. 767–769) to minimize stochasticity by adding perturbations to a pair of helical coils. Although only vacuum magnetic fields and an analytical magnetic field model are considered in this work, the adjoint calculation of the island width gradient could also be applied to a magnetohydrodynamic (MHD) equilibrium if the derivative of the magnetic field, with respect to the equilibrium parameters, is known. Using the island width gradient calculation presented here, more general gradient-based optimization methods can be applied to design stellarators with small magnetic islands. Moreover, the sensitivity of the island size may itself be optimized to ensure that coil tolerances, with respect to island size, are kept as high as possible.

Physics↗

Assessment of errors in analytic modeling of permanent magnet electron spectrometers for laser-plasma accelerators

Here, we assess the error in three treatments of a critical electron beam diagnostic for laser-plasma acceleration (LPA) experiments: a permanent magnet electron spectrometer. Since LPA electron spectrometers are often difficult to calibrate due to the scarcity of well-characterized, tunable electron beam sources in the appropriate energy range and the mechanical complexity of electron spectrometers, the standard of calibration is Hall probe measurements. We first compare the electron spectrometer performance between SIMION calculations and the Hall probe measurements. We find up to a 60% (<8%) error in determining the absolute energy for electrons below (above) 80 MeV when using the SIMION-modeled magnetic field vs the measured magnetic field due to SIMION overpredicting fringe field strength. The difference in spectrometer energy resolution is within ∼10% between the modeled and measured fields. We then assess a simple block model, which is commonly used. We find that the simple block model presented here sufficiently captures the predicted as-built spectrometer performance for any application provided that the user can tolerate uncertainties on absolute energy determination up to 5% and errors on energy resolution up to 1%.

Electromagnetic optics↗

Acceptance Testing of the Scorpius Sweep Magnet

A beam-dump magnet for Scorpius has been defined by LANL and designed and built by Radiabeam Technologies. Initial verification of the Radiabeam design used a magnetic field model provided by Radiabeam and analyzed by LANL tools during September 2021. Simulations indicated that the design was sufficient to meet deflection requirements to protect sensitive equipment in the case of a failed target. In January 2022, the magnet was fabricated and Radiabeam conducted measurements of the magnetic field as a function of position. The measurements deviate somewhat from the simulation. In particular, the peak magnetic field is 10-15% smaller in magnitude than expected from simulation. This note compares the magnetic field measurements to simulation and to measurements of the existing DARHT magnet, which was designed for the same purpose. We show a comparison of the measured beam deflection performance between Scorpius and DARHT and show updated results of LANL’s electron beam tracking code using the Scorpius measurements. Using field measurements we find that the DARHT magnet will deflect a 19.6 MeV electron beam by 46.6° and the Scorpius magnet will deflect by 44.6°. These results show that the as-built Scorpius magnet has met design requirements for beam deflection.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

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↗

Solar Energetic Proton Access to the Inner Magnetosphere During the September 7–8, 2017 Event

Here, the access of solar energetic protons into the inner magnetosphere on September 7–8, 2017 is investigated by following reversed proton trajectories to compute the proton cutoff energy using the Dartmouth geomagnetic cutoff code (Kress et al., 2010, https://doi.org/10.1029/2009sw000488). The cutoff energies for protons coming from the west and east direction, the minimum and maximum cutoff energy respectively, are calculated every 5 min along the orbit of Van Allen Probes using TS07 and the Lyon-Fedder-Mobarry (LFM) MHD magnetic field model. The result shows that the cutoff energy increases significantly as the radial distance decreases, and that the cutoff energy decreases with the building up of the ring current during magnetic storms. Solar wind dynamic pressure also affects cutoff suppression (Kress et al., 2004, https://doi.org/10.1029/2003gl018599). The LFM-RCM model shows stronger suppression of cutoff energy than TS07 during strong solar wind driving conditions. The simulation result is compared with proton flux measurements, showing consistent variation of the cutoff location during the September 7–8, 2017 geomagnetic storm.

79 ASTRONOMY AND ASTROPHYSICS↗

Quantifying the Effect of Magnetic Field Line Curvature Scattering on the Loss of Ring Current Ions

During geomagnetic storms, the ring current ions sometimes exhibit rapid loss as suggested by the fast recovery of the Dst index on a time scale of a few hours. Here, the effects of magnetic field line curvature (FLC) scattering on the loss of ring current ions, which have not been well quantified, are studied here by test particle simulations under the T89c magnetic field model. Our simulation results show that the prediction of ion loss based on a single–value cutoff of the κ parameter or maximum of δμ/μ of a single FLC scattering is not accurate. Instead, the e–folding lifetime (τ) for the loss of ring current ions due to cumulative FLC scattering has been calculated for different initial ion energies, equatorial pitch angles, and L shells under different geomagnetic conditions. The results show that in general the FLC scattering loss is faster for ions of higher energy, higher mass, smaller pitch angle, higher L, and at high Kp level. Specifically, we find that at Kp = 6 the lifetime can be <10 h at L > 5 for 100 s keV protons and at L > 4 for 100 s keV O + , which demonstrates that FLC scattering can be an important mechanism for the observed fast loss (τ < 10 h) of ring current ions during geomagnetic storms. Furthermore, we formulate an empirical formula for τ as a function of ion energy, pitch angle, position, species, and Kp. The empirical formula can be directly included in ring current models to account for the FLC scattering effects.

79 ASTRONOMY AND ASTROPHYSICS↗

Propagation of Chorus Waves Generated in Minimum‐B Pockets

Abstract We present the first ray tracing results of chorus waves generated in minimum‐B pockets. A new ray tracer is developed to accommodate an arbitrary magnetic field model, which enables us to construct minimum‐B pocket geometries using the Tsyganenko (1989) model. Rays with different frequencies and initial wave normal angles are launched from minimum‐B pockets and traced under different geomagnetic activity and hot electron density conditions. Our results indicate that chorus waves generated from minimum‐B pockets are usually highly localized and thus are unlikely to be detected unless the spacecraft is very close to the source region. These waves are also very unlikely to propagate into the plasmasphere. The propagation distances generally decrease with increasing geomagnetic activity.

Kang, Ning↗