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NASA NTRS2016

Electric Potential and Electric Field Imaging with Applications

The technology and techniques for remote quantitative imaging of electrostatic potentials and electrostatic fields in and around objects and in free space is presented. Electric field imaging (EFI) technology may be applied to characterize intrinsic or existing electric potentials and electric fields, or an externally generated electrostatic field may be used for (illuminating) volumes to be inspected with EFI. The baseline sensor technology, electric field sensor (e-sensor), and its construction, optional electric field generation (quasistatic generator), and current e-sensor enhancements (ephemeral e-sensor) are discussed. Demonstrations for structural, electronic, human, and memory applications are shown. This new EFI capability is demonstrated to reveal characterization of electric charge distribution, creating a new field of study that embraces areas of interest including electrostatic discharge mitigation, crime scene forensics, design and materials selection for advanced sensors, dielectric morphology of structures, inspection of containers, inspection for hidden objects, tether integrity, organic molecular memory, and medical diagnostic and treatment efficacy applications such as cardiac polarization wave propagation and electromyography imaging.

Generazio, Ed↗

NASA NTRS2015

Electric Field Generation in Martian Dust Devils

Terrestrial dust devils are known to generate electric fields from the vertical separation of charged dust particles. The particles present within the dust devils on Mars may also be subject to similar charging processes and so likely contribute to electric field generation there as well. However, to date, no Marsin situ instrumentation has been deployed to measure electric field strength. In order to explore the electric environment of dust devils on Mars, the triboelectric dust charging physics from the MacroscopicTriboelectric Simulation (MTS) code has been coupled to the Mars Regional Atmospheric ModelingSystem (MRAMS). Using this model, we examine how macroscopic electric fields are generated within martian dust disturbances and attempt to quantify the time evolution of the electrodynamical system.Electric fields peak for several minutes within the dust devil simulations. The magnitude of the electric field is a strong function of the size of the particles present, the average charge on the particles and the number of particles lifted. Varying these parameters results in peak electric fields between tens of millivolts per meter and tens of kilovolts per meter.

Aeolian processes↗

NASA NTRS2017

The Mars Crustal Magnetic Field Control of Plasma Boundary Locations and Atmospheric Loss: MHD Prediction and Comparison with MAVEN

We present results from a global Mars time-dependent MHD simulation under constant solar wind and solar radiation impact considering inherent magnetic field variations due to continuous planetary rotation. We calculate the 3-D shapes and locations of the bow shock (BS) and the induced magnetospheric boundary (IMB) and then examine their dynamic changes with time. We develop a physics-based, empirical algorithm to effectively summarize the multidimensional crustal field distribution. It is found that by organizing the model results using this new approach, the Mars crustal field shows a clear, significant influence on both the IMB and the BS. Specifically, quantitative relationships have been established between the field distribution and the mean boundary distances and the cross-section areas in the terminator plane for both of the boundaries. The model-predicted relationships are further verified by the observations from the NASA Mars Atmosphere and Volatile EvolutioN (MAVEN) mission. Our analysis shows that the boundaries are collectively affected by the global crustal field distribution, which, however, cannot be simply parameterized by a local parameter like the widely used subsolar longitude. Our calculations show that the variability of the intrinsic crustal field distribution in Mars-centered Solar Orbital itself may account for approx.60% of the variation in total atmospheric loss, when external drivers are static. It is found that the crustal field has not only a shielding effect for atmospheric loss but also an escape-fostering effect by positively affecting the transterminator ion flow cross-section area.

Global↗

NASA NTRS2017

The Martian Dust Devil Electron Avalanche: Laboratory Measurements of the E-Field Fortifying Effects of Dust-Electron Absorption

Analogous to terrestrial dust devils, charged dust in Mars dust devils should become vertically stratified in the convective features, creating large scale E-fields. This E-field in a Martian-like atmosphere has been shown to stimulate the development of a Townsend discharge (electron avalanche) that acts to dissipate charge in regions where charge build-up occurs. While the stratification of the charged dust is a source of the electrical energy, the uncharged particulates in the dust population may absorb a portion of these avalanching electrons, thereby inhibiting dissipation and leading to the development of anomalously large E-field values. We performed a laboratory study that does indeed show the presence of enhanced E-field strengths between an anode and cathode when dust-absorbing filaments (acting as particulates) are placed in the avalanching electron flow. Further, the E-field threshold condition to create an impulsive spark discharge increases to larger values as more filaments are placed between the anode and cathode. We conclude that the spatially separated charged dust creates the charge centers and E-fields in a dust devil, but the under-charged portion of the population acts to reduce Townsend electron dissipation currents, further fortifying the development of larger-than-expected E-fields.

Farrell, W. M.↗

NASA NTRS2018

Violation of Field Line Conservation and Associated Spatial Scales in Particle-In-Cell Simulations and MMS Data

For the first time, space flight technology exists to detect, in situ, violation of magnetic field line conservation. The violation of magnetic line conservation on scales smaller than the system size is a necessary and sufficient condition for finite magnetic field reconnection. We demonstrate that violation of line conservation produces a detectable, structured signature both in particle-in-cell simulations of reconnection and in data from the Magnetospheric Multi-Scale mission. In particle-in-cell simulations of asymmetric reconnection, the quantity-which we call M-that identifies this violation achieves significant values in electron skin depth-scale layers that extend away from the separator, with higher values emerging on the low-density, high-magnetic-field side of the current sheet. At the separator, M owes largely to perpendicular gradients in the parallel electric field, while it attains its highest values away from the separator in dispersed, layered structures associated with gradients in the perpendicular nonideal electric field and electron transport. Sub-ion scale bipolar forms of the quantity also appear further from the separator. In two MMS burst data intervals detecting the electron diffusion region, we find that M exceeds measurement uncertainties both at the separator and near the separatrices. One interval has highly sheared reconnecting fields and the other a stronger guide field. For one event, we determine the location and scale of M and the inner electron diffusion region relative to electron outflows and the magnetic separatrices. The measure can therefore serve as a potent diagnostic for magnetic reconnection in space measurements.

Wendel, Deirdre E.↗

NASA NTRS2018

A New Model of Jupiter's Magnetic Field From Juno's First Nine Orbits

A spherical harmonic model of the magnetic field of Jupiter is obtained from vector magnetic field observations acquired by the Juno spacecraft during its first nine polar orbits about the planet. Observations acquired during eight of these orbits provide the first truly global coverage of Jupiter's magnetic field with a coarse longitudinal separation of ~45 deg between perijoves. The magnetic field is represented with a degree 20 spherical harmonic model for the planetary ("internal") field, combined with a simple model of the magnetodisc for the field ("external") due to distributed magnetospheric currents. Partial solution of the underdetermined inverse problem using generalized inverse techniques yields a model ("Juno Reference Model through Perijove 9") of the planetary magnetic field with spherical harmonic coefficients well determined through degree and order 10, providing the first detailed view of a planetary dynamo beyond Earth.

Connerney, J. E. P.↗

NASA NTRS2018

Magnetic Reconnection on Dayside Crustal Magnetic Fields at Mars: MAVEN Observations

The identification of magnetic reconnection on the dayside of Mars has been elusive owing to the lack of comprehensive plasma and field measurements. Here we present direct measurements of dayside in situ reconnection signatures by the comprehensive particles and fields package on board the Mars Atmosphere and Volatile EvolutioN (MAVEN) spacecraft over strong crustal magnetic fields in the southern hemisphere of Mars. During a crossing of a bifurcated current sheet consisting of northward and southward magnetic fields, MAVEN recorded (i) ionospheric photoelectrons trapped on closed magneticfield lines, (ii) Hall magnetic fields and a nonzero normal field with polarity consistent with a crossing northward of the X line, and (iii) northward Alfvenic ion jets. Dayside magnetic reconnection on crustal magnetic fields could control the global configuration and topology of the Martian magnetosphere and alter the ion escape pattern from the dayside ionosphere.

Magnetic↗

NASA NTRS2018

Overview of Methods for Deriving the Radiative Transfer Theory from the Maxwell Equations. I: Approach Based on the Far-Field Foldy Equations

In this paper, we revisit, with further enhancements and clarifications, the self-consistent first-principles approach developed previously for deriving the vector radiative transfer theory for a discrete random medium with a sparse concentration of particles. We specifically consider the case of a plane-parallel particulate layer embedded in an otherwise homogeneous unbounded medium. The solution method is based on the far-field Foldy equations, an order-of-scattering expansion for the total field derived under the Twersky approximation, the computation of the coherent field by assuming that the positions of the particles are uncorrelated, and the ladder approximation for the coherency dyadic. The latter yields an integral equation for the diffuse specific coherency dyadic, defined through an angular spectrum repre- sentation for the coherency dyadic, which in turn, gives the vector radiative transfer equation for the diffuse specific intensity column vector. We analyze specifically the computation of the coherent field for inhomogeneous particulate media and multiple species of particles, the continuous extension of the far-field representation to the near field, the Foldy approximation, and the Foldy integral equation for the coherent field. Finally, we discuss the transition from the vector to the scalar radiative transfer equation.

Doicu, Adrian↗

NASA NTRS2018

Field Camp for Astronauts: NASA's Geoscience Training Program for Planetary Exploration

Fifty years ago Apollo astronauts walked on the Moon to explore the geology and collect samples for Earth return. Several authors have discussed the strategic planning and training that enabled the Apollo successes, and assembled recommendations regarding today’s lunar science objectives and astronaut training required to achieve those science goals. Since the 1980s, geoscience training for astronauts focused on observing the Earth from orbit. Today, we are building a geoscience training program to support informed Earth observations as well as the exploration culture for future human missions to the Moon and Mars. Our team partnered with JSC’s crew training and astronaut offices to develop our 4-week geoscience program for the 2017 astronaut class. Because the astronauts have a variety of professional backgrounds, we provide a broad introduction to Earth and planetary sciences. But our prime focus is 2 weeks of intensive field work, a methodology introduced with the 2013 astronaut class. We completed the first half of the training – a field trip to observe hurricane deposits along Galveston Bay; keynotes by Apollo colleagues highlighting Apollo experiences; a tightly-integrated week of introductory geology in the classroom followed by a week of fieldwork in the Rio Grande del Norte National Monument. The classroom included interactive map exercises that allowed the students to progressively build a base map of the field area that they used as a starting point for their week-long mapping exercise. We divided the class into small mapping groups to conduct their observations, mapping and interpretation of the geology. In addition to learning geological field work, our field training provided the platform for practicing expeditionary leadership, a key skill set valued by NASA for astronaut crews. Next summer the capstone fieldwork for the 2017 astronauts will include both mapping and rock sampling. Throughout the mapping, the class will collect additional data to help inform field and sampling decisions using diagnostic field instruments that are being tested in analog settings for their operational efficacy for future planetary exploration.

Evans, Cynthia A.↗

NASA NTRS2018

Predicting Global Ground Geoelectric Field With Coupled Geospace and Three-Dimensional Geomagnetic Induction Models

We forecast the global effects of space weather on the geoelectric and geomagnetic fields using a novel combination of methods. We use a realistic three-dimensional (3-D) model of Earth's electrical conductivity and a realistic representation of magnetospheric and ionospheric current systems. Our scheme involves the following steps: (1) We run a global magnetohydrodynamic model of the magnetosphere coupled to an electrostatic model of the ionosphere. (2) We calculate a global time series of the ground magnetic field resulting from the ionospheric, field-aligned, and magnetospheric currents of the global magnetohydrodynamic model. (3) We approximate this external field by an equivalent source current flowing in a thin shell above Earth. (4) We calculate a global time series of geoelectric and geomagnetic fields from the equivalent current and a 3-D conductivity model of Earth that also takes into account the coast effect due to large horizontal conductivity gradient. We verify our implementation by comparing the results against known analytic and numeric solutions, and then apply our scheme to the geomagnetic storm of 14 and 15 December 2006. In particular, we show that accounting for 3-D structure of Earth's conductivity results in significantly enhanced geoelectric field at large lateral gradients of conductivity, especially in coastal regions, both at middle and high latitudes. In the studied geomagnetic storm the largest values of 3-D geoelectric field are detected at high latitudes reaching 2.5 volts per kilometer and the 3-D effect extends inland by a few hundred kilometers.

Honkonen, I.↗

NASA NTRS2019

Unraveling the Internal Magnetic Field Structure of the Earth-Directed Interplanetary Coronal Mass Ejections During 1995-2015

The magnetic field configurations associated with interplanetary coronal mass ejections (ICMEs) are the in situ manifestations of the entrained magnetic structure associated with coronal mass ejections (CMEs). We present a comprehensive study of the internal magnetic field configurations of ICMEs observed at 1 AU by the Wind mission during 1995-2015. The goal is to unravel the internal magnetic structure associated with the ICMEs and establish the signatures that validate a flux-rope structure. We examine the expected magnetic field signatures by simulating spacecraft trajectories within a simple flux rope, i.e., with circular–cylindrical (CC) helical magnetic field geometry. By comparing the synthetic configurations with the 353 ICME in situ observations, we find that only 152 events ( Fr ) display the clear signatures of an expected axial-symmetric flux rope. Two more populations exhibit possible signatures of flux rope; 58 cases ( F− ) display a small rotation ( <90° ) of the magnetic field direction, interpreted as a large separation of the spacecraft from the center, and, 62 cases ( F+ ) exhibit larger rotations, possibly arising from more complex configuration. The categories, Cx (14%) and E events (9%), reveal signatures of complexity possibly related with evolutionary processes. We then reconstruct the flux ropes assuming CC geometry. We examine the orientation and geometrical properties during the solar activity levels at the end of Solar Cycle 22 (SC22), SC23 and part of SC24. The orientation exhibits solar cycle trends and follow the heliospheric current sheet orientation. We confirm previous studies that found a Hale cycle dependence of the poloidal field reversal. By comparing our results with the occurrence of CMEs with large angular width ( AW>60∘ ) we find a broad correlation suggesting that such events are highly inclined CMEs. The solar cycle distribution of bipolar vs. unipolar Bz configuration confirms that the CMEs may remove solar cycle magnetic field and helicity.

Nieves-Chinchilla, Teresa↗

NASA NTRS2019

HERMeS Thruster Magnetic Field Topology Optimization Study: Performance, Stability, and Wear Results

NASA's Hall Effect Rocket with Magnetic Shielding (HERMeS) 12.5kW Technology Demonstration Unit-1 (TDU-1) has been the subject of extensive technology maturation in preparation for flight system development. The TDU-1 thruster implements a magnetically shielded field topology and has demonstrated the elimination of the discharge channel erosion as a life limiting mechanism. Extensive wear testing the TDU Hall thrusters has identified the thruster front pole covers as the next life limiting component. This effort aims to explore and investigate alternate magnetic field topologies to assess whether reductions in the front pole cover erosion can be attained while still maintaining very low erosion rates on the discharge channel walls. Four candidate magnetic field topologies that reduce the effectiveness of the shielding along the discharge channel walls with the intent to also reduce the erosion rates along the front pole covers were designed. Three of the four candidate magnetic field topologies (B1, B2, and B4) have been manufactured and were subjected to an extensive test campaign that includes laser induced fluorescence (LIF), performance, stability, wear, plume, and thermal characterization. In Phase I, LIF measurements along the discharge chamber centerline found that upstream retraction of the thruster's peak magnetic field does result in an upstream shift of the acceleration zone but the magnitude of the shift does not correspond one-to-one to the shift in the location of the peak radial magnetic field magnitude. Phase II test segment results found at a normalized thruster magnetic field setting of 1, the thruster performance was very similar for all configurations. Discharge current waveforms ripple data indicated that configurations Bo, B1, and B2 has similar oscillatory profiles with the B2 configuration transitioning to higher oscillatory mode at 400V instead of the 450V observed for configurations B0 and B1. Configuration B4 ripple data indicates that the thruster was operating in a very oscillatory mode above 350V. Inner front pole cover erosion rates for configuration B1 were approximately 65% relative to B0 and the erosion rates for configuration B2 were 40% relative to B0. Analysis of the performance, stability, and wear test results of configurations B0, B1, B2, and B4 indicates that configuration B2 may present an alternative option for B0 because it mostly maintained the same performance and stability as configuration B0 but with 40% of the inner front pole cover erosion rate.

Kamhawi, Hani↗

NASA NTRS2020

Helicity in the Large-Scale Galactic Magnetic Field

We search for observational signatures of magnetic helicity in data from all-sky radio polarization surveys of the Milky Way Galaxy. Such a detection would help confirm the dynamo origin of the field and may provide new observational constraints for its shape. We compare our observational results to simulated observations for both a simple helical field, and for a more complex field that comes from a solution to the dynamo equation. Our simulated observations show that the large-scale helicity of a magnetic field is reflected in the large-scale structure of the fractional polarization derived from the observed synchrotron radiation and Faraday depth of the diffuse Galactic synchrotron emission. Comparing the models with the observations provides evidence for the presence of a quadrupolar magnetic field with a vertical component that is pointing away from the observer in both hemispheres of the Milky Way Galaxy. Since there is no reason to believe that the Galactic magnetic field is unusual when compared to other galaxies, this result provides further support for the dynamo origin of large-scale magnetic fields in galaxies.

J. L. West↗

NASA NTRS2021

SHARPEN: A Scheme to Restore the Distribution of Averaged Precipitation Fields

A key strategy in obtaining complete global coverage of high-resolution precipitation is to combine observations from multiple fields, such as the intermittent passive microwave observations, precipitation propagated in time using motion vectors, and geosynchronous infrared observations. These separate precipitation fields can be combined through weighted averaging, which produces estimates that are generally superior to the individual parent fields. However, the process of averaging changes the distribution of the precipitation values, leading to an increase in precipitating area and a decrease in the values of high precipitation rates, a phenomenon observed in IMERG. To mitigate this issue, we introduce a new scheme called SHARPEN (Scheme for Histogram Adjustment with Ranked Precipitation Estimates in the Neighborhood), which recovers the distribution of the averaged precipitation field based on the idea of quantile mapping applied to the local environment. When implemented in IMERG, precipitation estimates from SHARPEN exhibit a distribution that resembles that of the original instantaneous observations, with matching precipitating area and peak precipitation rates. Case studies demonstrate its improved ability in bridging between the parent precipitation fields. Evaluation against ground observations reveals a distinct improvement in precipitation detection skill, but also a slightly reduced correlation likely because of a sharper precipitation field. The increased computational demand of SHARPEN can be mitigated by striding over multiple grid boxes, which has only marginal impacts on the accuracy of the estimates. SHARPEN can be applied to any precipitation algorithm that produces an average from multiple input precipitation fields and is being considered for implementation in IMERG V07.

Jackson Tan↗

NASA NTRS2020

Conditions of Loss Cone Filling by Scattering on the Curved Field Lines for 30 keV Protons During Geomagnetic Storm as Inferred From Numerical Trajectory Tracing

The rate of pitch angle scattering on the curved magnetic field lines is well parameterized by the ratio of the minimum field line curvature radius to the maximum effective particle gyroradius (K = RC/rg). The critical value of this ratio (Kcr) corresponding to the loss cone filling is of special interest since it corresponds to the low altitude isotropic boundaries (IBs). The early theoretical estimates gave Kcr = 8, whereas recent estimations of the K parameter on the field lines corresponding to the observed IBs during the geomagnetic storms revealed KIB values in the range of 3–30. We numerically trace the trajectories of the 30 keV protons in the magnetic field of the global magnetohydrodynamic simulation of the intense storm in order to infer statistical distribution of Kcr. The electric field and effects of nonstationarity are neglected in this study. It is found that although the Kcr values do show some variations during the course of the storm, its range is rather narrow 4 < Kcr < 8. The result suggests that higher KIB values found in the observational studies, if not caused by the magnetosphere-ionosphere mapping error, should be attributed to some other mechanism of pitch angle scattering. The Kcr values tend to be lower (4–6) during the main phase because the region of low K values approaches the Earth and the equatorial loss cone size becomes larger due to a larger equatorial magnetic field in the near earth region. The remaining variation of Kcr is explained by the presence of the guide component of the magnetic field.

S Dubyagin↗

NASA NTRS2021

Lower-Hybrid Drift Waves and Their Interaction With Plasmas in A 3D Symmetric Reconnection Simulation With Zero Guide Field

We investigate lower-hybrid drift waves (LHDW) in symmetric magnetic reconnection with zero guide field using three-dimensional particle-in-cell simulations. The long-wavelength mode with develops in the bifurcated electron current layer around the X-line within the width of the electron meandering motion from the mid-plane, where is the ion (electron) gyroradius. The short-wavelength mode with develops in the separatrix region downstream of the electron outflow jet, producing electron vortices in the background flow frame. Electrons follow the E × B drift with corrections from the diamagnetic drift and are heated inside the vortices with diverging electric fields. In the vortices, ions have comparable E × B and inertia drifts, which together mostly cancel the diamagnetic drift. Toward the center of diverging field vortices, ions are decelerated, leading to a decrease in the perpendicular temperature, while the loss of low-energy ions results in an increase in the parallel temperature. Parallel electric fields exist as a combination of the LHDW wave field projected to the magnetic field direction and the penetration of whistler waves that are mainly outside of the LHDW layer. The magnetic flux tube is twisted in the vortices. The twist may potentially lead to slippage reconnection, as indicated by the non-uniform parallel potential variation across field lines, while the periodic variations of the twisting directions are a limiting factor.

Shan Wang↗

NASA NTRSDate not supplied

3-D Mapping of the Magnetic Field in the Active Region by the CLASP2.1 Rocket Experiment

Comprehensive magnetic field measurements of the solar atmosphere are crucial for understanding energy transport from the photosphere to the corona and its dissipation. However, observations of the magnetic field in the chromosphere and the upper atmospheric layers above the chromosphere, where the gas pressure dominance changes from (β>1) to magnetic pressure dominance (β<1), are overwhelmingly lacking. Therefore, we have focused on the polarization of ultraviolet radiation emitted from the upper chromosphere and transition layer and have conducted the Japan-U.S.-EU observation rocket experiment CLASP in order to demonstrate its usefulness. In CLASP2.1 conducted on October 8, 2021, scan observations were made at 16 locations in the active region, and Stokes (intensity $I$, linearly polarized $Q$, $U$, circularly polarized $V$) spectra in the 280~nm wavelength range were obtained. Looking at the observed region with AIA 171~{¥AA} on board the SDO satellite, we see that it consists of a region where a structure corresponding to the foot of a high-temperature loop called moss (moss) is seen, and a region where a low-temperature loop spreading from a sunspot is seen. We focused on the ionized magnesium $h$ & $k$ lines (emitted from the middle and uppermost of the chromosphere) and the manganese lines (emitted from the low part of the chromosphere), which show particularly prominent circular polarization, and derived the line-of-sight magnetic fields in the low, middle and uppermost parts of the chromosphere by applying weak field approximation to them. Furthermore, by combining the results with observations by the Solar Optical Telescope onboard the solar observing satellite HINODE, we obtained three-dimensional information on the magnetic field in the active region from the photosphere to the uppermost part of the chromosphere. In general, the magnetic field in the active region becomes weaker and smoother as one goes up in the sky, as reported by the CLASP2 observation (Ishikawa et al. 2021). However, in some regions, polarity reversal was observed only in the uppermost part of the chromosphere, and comparison with high spatial resolution transition layer and coronal images recorded by SDO/AIA revealed the connection between the magnetic field structure in the chromosphere and coronal loops.

R. Ishikawa↗

DOE OSTI2026

In-Situ Magnetic Field Reconstruction in the MAGIS-100 Experiment

Long-baseline atom interferometers such as the Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) require stringent control and continuous characterization of background magnetic fields and spatial gradients to prevent systemic phase shifts that mimic ultralight dark matter or gravitational wave signatures. Because direct sensor placement within the ultra-high vacuum beam pipe is infeasible, in-situ magnetic field monitoring relies on external sensor arrays situated in the surrounding annular region. This work demonstrates a field reconstruction framework for a 5.3-meter MAGIS-100 modular section using finite-element Opera simulations. Transverse magnetic fields are expanded using a cylindrical multipole framework as informed by Fermilab’s Muon g-2 experiment, with magnetometer array configurations optimized via Fisher information matrix D-optimality. Inverting external sensor readings through a Gauss-Newton scheme recovers interior tube fields across distinct axial positions. In the discontinuity-averse uniform region (slice pair P4), the model achieves sub-noise-floor performance with a cross-validated root-mean-square error (RMSE) of $6.7227 \times 10^{-4}\text{ A/m}$ ($0.845\times$ sensor noise floor) and an interior field coefficient of variation of $1.71\%$. An elbow criterion in the Fisher bounds establishes $n_{\text{max}} = 2$ as the optimal multipole truncation order to prevent noise amplification from over-parameterization, with $n_{\text{max}} = 3$ (sextupole) order chosen for analysis to demonstrate further complexity and cross-pair comparison. Furthermore, analytical differentiation of the fitted multipole coefficients yields dense spatial maps of the transverse Jacobian gradient matrix $\nabla \mathbf{H}$ along with propagated $1\sigma$ uncertainty bounds across the beam region ($r \le 2.75\text{ in}$). This operational framework confirms that external magnetometer arrays can reliably monitor magnetic field uniformity and spatial gradients along the 100-meter flight path given appropriate sampling for any complexity order.

Appleby, Darwin [William Rainey Harper Coll.; Ferm↗