Field Model Research Articles

The research on the applicability of different typhoon wind fields in the simulation of typhoon waves in China’s coastal waters

Typhoon waves possess significant destructive potential, and their numerical simulation relies on accurate sea surface wind fields. An evaluation of different combinations of the radial air pressure distribution coefficient B and the radius of maximum wind speed (Rmax) in the Holland wind field (HWF) model was conducted to determine the optimal configuration. The HWF and the ERA5 wind field (EWF) were used as input wind fields to drive the typhoon wave model for China’s coastal waters. Validation results indicated that neither wind field accurately reflected real conditions; therefore, a hybrid wind field (HBWF) was created by combining HWF and EWF using weighting coefficients that vary with the radius of wind speed to enhance accuracy. Simulation results showed that the HBWF improved the accuracy of significant wave heights (SWHs), with a mean relative error of 25.29%, compared to 32.48% for HWF and 27.94% for EWF. Additionally, HBWF also demonstrated the best performance in terms of root mean square error (RMSE) and consistency index. Overall, the HBWF enhances the simulation accuracy of typhoon waves in China's coastal waters.

Analysis of the air gap magnetic field in cylindrical magnetic couplings based on mathematical and finite element approach

The air gap magnetic field of a magnetic coupling plays a crucial role in the examination of its static torque and eddy current losses. Precise characterization of the air gap magnetic field is essential for ensuring the accuracy of performance assessments of the magnetic coupling. This study focuses on the cylindrical magnetic coupling as the subject of investigation, employing mathematical analysis and finite element methods to evaluate and quantify the magnetic field properties. The study establishes a mathematical model of the magnetic field of a magnetic coupling based on electromagnetic field principles and the superposition theorem. An analytical formula for the magnetic flux density distribution of the air-gap magnetic field is derived. Subsequently, a three-dimensional magnetic field finite element model is created using the finite element method to numerically calculate the air gap magnetic field and validate the analytical formula. The research also explores the periodic distribution characteristics of the magnetic field in the air gap, analyzing axial differences in magnetic flux density value and direction distribution influenced by end effects.

Finite element modeling of concentrated impact loads on the masticatory muscles at the head

AbstractNumerical simulation has great potential to provide a more comprehensive understanding of human impact response to injury mechanisms. Finite Element (FE) models are used as a tool to study human injuries in greater detail, for example, the THUMS (Total Human Model Safety of TOYOTA) model, which is widely used as a reliable human model in different fields to predict human injuries such as fractures, internal organ damage, and brain tissue injuries. However, no available FE model can be used to simulate human‐robot collisions based on standards ISO/TS 15066 and the biomechanical characteristics of human soft tissues in vivo. The authors have developed a head model based on the structures (dimensions and anatomy) of the THUMS head model, specifically designed to simulate impact loads on the masticatory muscles. Based on medical imaging (MRI) data, the soft tissues at the location of the masticatory muscles in the THUMS head are transformed from monolayer to multilayer, that is, a composite geometry of skin‐fat‐muscle each with its own material model and parameters. The model was optimized and validated using the experimental data from the Fraunhofer IFF subjects study, which determined biomechanical thresholds for specific body locations in ISO/TS 15066 under dynamic collisions.

Orientational Disorder of Alcohol Molecules at Their Solution Surfaces in Low Concentration Range: A Molecular Dynamics Simulation Study.

Molecular dynamics (MD) simulations of short-chain alcohols (methanol, ethanol, and 1-propanol) in solution were carried out to examine the orientational disordering (randomizing) of alcohol molecules at the surface under diluted conditions. Recent vibrational sum frequency generation (VSFG) spectroscopy, combined with photoelectron spectroscopy, has successfully measured the disordering structure at low concentrations. The present MD simulations accurately reproduce this observation for the first time. To ensure reliable results through MD simulations, several widely used force field models for water and alcohol, including polarizable models, were examined. This examination involved a comparison of structural and thermodynamic quantities, such as surface density times orientation and surface excess concentration, which were obtained through surface-specific measurements like VSFG spectroscopy and surface tension measurements. It is found that the width of the density profile for alcohol molecules at the surface, along the surface normal, increases as the concentration decreases in the diluted condition, which is consistent with the results obtained from the previous neutron and X-ray grazing incidence reflection experiments. A molecular mechanism explaining the disordering of alcohol molecules with decreasing concentration is also discussed.

Investigation of laser sintering process parameters for anode foils in aluminium electrolytic capacitors considering temperature distribution

Abstract The anode foil is a critical component of aluminium electrolytic capacitors, with its performance directly impacting the overall quality of the capacitors. Currently, sintered anode foil with excellent bending resistance and high specific capacitance is considered an ideal material for capacitor manufacturing; however, research on its optimal sintering parameters remains insufficient. In this study, a three-dimensional temperature field model is developed within the Comsol Multiphysics (6.0) environment, accounting for the temperature dependence of aluminium. By varying laser power and scanning speed, the temperature distribution along the laser scanning trajectory is determined, facilitating the identification of optimal process parameters for laser sintering anode foils in electrolytic capacitors. Subsequent laser sintering experiments validate the accuracy of these parameters. The findings indicate that the peak temperature of the molten pool rises with increased laser power and decreased scanning speed. The optimal process parameters for laser sintering anode foils in electrolytic capacitors are a powder layer thickness of 50 µm, a laser power of 140 W, and a scanning speed of 0.05 m/s. The specific capacitance of laser-sintered anode foil, formed at voltages of 375 V and 520 V, ranges from 0.847 to 1.157 μF/cm² and 0.717 to 0.935 μF/cm², respectively, when the particle size is between 3 and 4 μm. A specific capacitance of 0.733 μF/cm² can be achieved, which meets the performance requirements for aluminium electrolytic capacitors.

One- and two-photon excitation dynamics using semiclassical electron force field model.

We have extended the semiclassical-based electron force-field simulation by introducing field-electron interaction to enable us to describe linear and nonlinear electronic excitation dynamics of a condensed matter system with low computational cost. To verify the simulation method, as a first step, numerical examples of interaction dynamics of simple systems (H atom, SiH4 molecule, and Si crystalline solid) with applied short electric field pulse as well as the obtained absorbed energies by the one- and two-photon excitations have been reported along with comparison with quantum dynamics calculations as reference.

Weighted Fusion Method of Marine Gravity Field Model Based on Water Depth Segmentation

Among the marine gravity field models derived from satellite altimetry, the Scripps Institution of Oceanography (SIO) series and Denmark Technical University (DTU) series models are the most representative and are often used to integrate global gravity field models, which were inverted by the deflection of vertical method and sea surface height method, respectively. The fusion method based on the offshore distance used in the EGM2008 model is just model stitching, which cannot realize the true fusion of the two types of marine gravity field models. In the paper, a new fusion method based on water depth segmentation is proposed, which established the Precision–Depth relationship of each model in each water depth segment in the investigated area, then constructed the FUSION model by weighted fusion based on the precision predicted from the Precision–Depth relationship at each grid in the whole region. The application in the South China Sea shows that the FUSION model built by the new fusion method has better accuracy than SIO28 and DTU17, especially in shallow water and offshore areas. Within 20 km offshore, the RMS of the FUSION model is 5.10 mGal, which is 8% and 4% better than original models, respectively. Within 100 m of shallow water, the accuracy of the FUSION model is 4.01 mGal, which is 14% and 12% higher than the original models, respectively. A further analysis shows that the fusion model is in better agreement with the seabed topography than original models. The new fusion method can blend the effective information of original models to provide a higher-precision marine gravity field.

Precise Geoid Determination in the Eastern Swiss Alps Using Geodetic Astronomy and GNSS/Leveling Methods

Astrogeodetic deflections of the vertical (DoVs) are close indicators of the slope of the geoid. Thus, DoVs observed along horizontal profiles may be integrated to create geoid undulation profiles. In this study, we collected DoV data in the Eastern Swiss Alps using a Swiss Digital Zenith Camera, the COmpact DIgital Astrometric Camera (CODIAC), and two total station-based QDaedalus systems. In the mountainous terrain of the Eastern Swiss Alps, the geoid profile was established at 15 benchmarks over a two-week period in June 2021. The elevation along the profile ranges from 1185 to 1800 m, with benchmark spacing ranging from 0.55 km to 2.10 km. The DoV, gravity, GNSS, and levelling measurements were conducted on these 15 benchmarks. The collected gravity data were primarily used for corrections of the DoV-based geoid profiles, accounting for variations in station height and the geoid-quasigeoid separation. The GNSS/levelling and DoV data were both used to compute geoid heights. These geoid heights are compared with the Swiss Geoid Model 2004 (CHGeo2004) and two global gravity field models (EGM2008 and XGM2019e). Our study demonstrates that absolute geoid heights derived from GNSS/leveling data achieve centimeter-level accuracy, underscoring the precision of this method. Comparisons with CHGeo2004 predictions reveal a strong correlation, closely aligning with both GNSS/leveling and DoV-derived results. Additionally, the differential geoid height analysis highlights localized variations in the geoid surface, further validating the robustness of CHGeo2004 in capturing fine-scale geoid heights. These findings confirm the reliability of both absolute and differential geoid height calculations for precise geoid modeling in complex mountainous terrains.

New Rankine Vortex Models Developed Based on SMAP Measurements

Abstract The L-band passive microwave radiometer on board the NASA Soil Moisture Active Passive (SMAP) satellite can measure brightness temperature to retrieve sea surface wind speed under tropical cyclone (TC) conditions without being affected by rainfall or signal saturation caused by high wind speeds. Based on this advantage, this paper used the SMAP wind products for parameterizing the key decay exponent α of the Rankine vortex model (a traditional parametric model of the TC wind field) and finally developed new Rankine models. The SMAP dataset included 67 TC cases. Through data statistics, we examined the relationship between α and the maximum wind speed Um, the relationship between α and the radius of maximum wind speed (Rm), and the relationship between Rm and the averaged radius of 17 m s−1 (R17). Results showed that the three relationships were both positive correlations, indicating that α can be parameterized in three empirical ways. The first way is to calculate solely with Um. The second way is to calculate solely with Rm. The third way is to calculate Um and Rm together. The three ways correspond to three new models: the SMAP Rankine Model-1 (SRM-1), the SMAP Rankine Model-2 (SRM-2), and the SMAP Rankine Model-3 (SRM-3). Finally, comparisons were made between the new models and three existing Rankine models, according to the model simulations and the Advanced Microwave Scanning Radiometer 2 measurements of 49 TC cases. Results showed that the SRM-3 performed best overall. Significance Statement Ocean surface wind fields are important driving factors for numerical models to guide evolution forecasting and risk assessment of tropical cyclones. The purpose of this study is to utilize the SMAP products to modify the traditional Rankine vortex model for tropical cyclone surface wind field estimation. Rankine decay exponent was found between 0.3 and 0.9 and positively dependent upon maximum wind speed and its radius. Based on these characteristics, the proposed new Rankine models could calculate the decay exponent and further the TC surface wind field symmetrically with high accuracy, simply using maximum wind and its radius. This paper shows a practical use of SMAP measurements on TC wind field monitoring, analyzing, and modeling.

Wind Turbine Gearbox Anomaly Detection Using Signal-to-Image Processing Algorithms and Convolutional Autoencoder

Abstract In this study, signal-to-image conversion techniques coupled with a convolutional auto encoder (CAE) are used for the detection of anomalies in the wind turbine (WT) gearbox system. Firstly, the time series data is converted to images using six different algorithms. Thereafter, these images are stacked into the multi-dimensional structures known as “data cubes”, which is finally fed into CAE for the anomaly identification. The results of this study demonstrate the enhanced efficacy of the method specially in the Gramian Angular Field model in detecting anomalies accurately, suggesting a viable path towards the implementation of dependable and affordable WT monitoring systems. This will open the door for the renewable energy industry’s condition monitoring procedures to become more automated and digitalized.

Creep-thermal fatigue behavior of thin-walled structures with holes and a creep-thermal fatigue-oxidation phase field model

An experiment was conducted to evaluate the creep-thermal fatigue (CTF) behavior of thin-walled structures with holes. To achieve this, a high-temperature hold phase was added in the testing. The crack propagation of CTF is driven by the combined effects of creep, thermal fatigue, and oxidation. Therefore, a creep-thermal fatigue-oxidation phase field model was developed to simulate CTF behavior. The model accounts for the interaction between creep damage and fatigue damage, as well as the oxidation effect. A creep degradation function was formulated based on classical damage theory, and two creep damage models were compared. Two physically meaningful strategies were proposed to describe oxidation-induced fatigue damage. Finally, the applicability of model to creep-fatigue was validated.

Conformably Variable Geocentric Axial Dipole at ca. 2.1 Ga: Paleomagnetic Dispersion of the Indin Dyke Swarm, Slave Craton

AbstractPrecambrian paleomagnetic studies are critical for testing paleogeographic reconstructions in deep time but rely on the fidelity of the assumption of the geocentric axial dipole (GAD) hypothesis. With high‐reliability data from mafic dykes and volcanic rocks, the scatter of individual virtual geomagnetic poles (VGPs) can be used to test simple GAD models. In order to conduct such a test, the VGPs must be adequate in number and in spatial coverage of the sampling sites. In this study, we targeted the 2.1 Ga Indin dyke swarm of the Slave craton. Building on previous sampling of the Indin dyke swarm in the western and central parts of southern Slave craton, we report results from 9 additional sites in the central and eastern parts of the craton, sites that significantly expand the width of the dyke swarm across the entire craton. The VGPs obtained from 7 of 9 newly identified Indin dykes are broadly similar to previously reported directions, expanding the total of VGPs for individual Indin dykes to n = 28, which is sufficient for a test of the GAD‐based statistical models using VGP scatter. The high VGP scatter of the Indin swarm can be attributed to the relatively high paleolatitude of 56° ± 6° for the Slave craton at the time of dyke emplacement. The Indin data have VGP scatter that is consistent with field models associated with the GAD hypothesis for the indicated paleolatitude, thus confirming the fidelity of the GAD field at ca. 2.1 Ga.

Ultraviolet Radiation Fields in Star-forming Disk Galaxies: Numerical Simulations with TIGRESS-NCR

With numerical simulations that employ adaptive ray-tracing (ART) for radiative transfer at the same time as evolving gas magnetohydrodynamics, thermodynamics, and photochemistry, it is possible to obtain a high-resolution view of ultraviolet (UV) fields and their effects in realistic models of the multiphase interstellar medium. Here, we analyze results from TIGRESS-NCR simulations, which follow both far-UV (FUV) wavelengths, important for photoelectric heating and polycyclic aromatic hydrocarbon excitation, and the Lyman continuum (LyC), which photoionizes hydrogen. Considering two models, representing solar neighborhood and inner-galaxy conditions, we characterize the spatial distribution and time variation of UV radiation fields, and quantify their correlations with gas. We compare four approximate models for the FUV to simulated values to evaluate alternatives when full ART is infeasible. By convolving FUV radiation with density, we produce mock maps of dust emission. We introduce a method to calibrate mid-IR observations, for example from JWST, to obtain high-resolution gas surface density maps. We then consider the LyC radiation field, finding most of the gas exposed to this radiation to be in ionization–recombination equilibrium and to have a low neutral fraction. Additionally, we characterize the ionization parameter as a function of the environment. Using a simplified model of the LyC radiation field, we produce synthetic maps of emission measure (EM). We show that the simplified model can be used to extract an estimate of the neutral fraction of the photoionized gas and mean free path of ionizing radiation from observed EM maps in galaxies.

에듀테크를 활용한 자기주도적 음악 창작 영역의 지도 방안 연구

The 2022 revised curriculum was announced to help students develop core competencies required by the future society and grow into leading individuals in line with the changing times. The purpose of this study is to examine the utilization of various Edtech programs and effective teaching methods in the field of Edtech through them. We believe that what students need to live in a rapidly changing future society is self-directedness to make judgments and decisions on their own and the ability to utilize digital media, and we plan to utilize Edtech as a method to develop these abilities. In this study, we analyzed the 2022 revised music curriculum and examined teaching methods in the field of music creation using Edtech. The most noticeable change in the revised music curriculum is the establishment of a creative field, and since there are many ways to apply Edtech to the creative field, we focused on creative activities. By analyzing the revised curriculum, we organized the activity topics of creative classes, explored Edtech that can be utilized in music classes, and then presented learning strategies that can enhance self-directedness. Based on this, a teaching and learning model for the music creation field utilizing Edtech was presented and a teaching and learning guidance plan was proposed accordingly. Edtech can be utilized in various ways in music creation classes and can be said to be an essential element in education for the future. Through this study, it was confirmed that the use of Edtech in music and creation classes can foster learners' self-directed capabilities and digital media utilization capabilities. It is hoped that this study will be helpful to teachers in the field and that follow-up research on this topic will continue.

Research on Vehicle Risk Field Model for Edge Infrastructure at Cooperative Driving

Research on Vehicle Risk Field Model for Edge Infrastructure at Cooperative Driving

A Simulation Study on the Effect of Supersonic Ultrasonic Acoustic Streaming on Solidification Dendrite Growth Behavior During Laser Cladding Based on Boundary Coupling

Laser cladding has unique technical advantages, such as precise heat input control, excellent coating properties, and local selective cladding for complex shape parts, which is a vital branch of surface engineering. During the laser cladding process, the parts are subjected to extreme thermal gradients, leading to the formation of micro-defects such as cracks, pores, and segregation. These defects compromise the serviceability of the components. Ultrasonic vibration can produce thermal, mechanical, cavitation, and acoustic flow effects in the melt pool, which can comprehensively affect the formation and evolution for the microstructure of the melt pool and reduce the microscopic defects of the cladding layer. In this paper, the coupling model of temperature and flow field for the laser cladding of 45 steel 316L was established. The transient evolution laws of temperature and flow field under ultrasonic vibration were revealed from a macroscopic point of view. Based on the phase field method, a numerical model of dendrite growth during laser cladding solidification under ultrasonic vibration was established. The mechanism of the effect of ultrasonic vibration on the solidification dendrite growth during laser cladding was revealed on a mesoscopic scale. Based on the microstructure evolution model of the paste region in the scanning direction of the cladding pool, the effects of a static flow field and acoustic flow on dendrite growth were investigated. The results show that the melt flow changes the heat and mass transfer behaviors at the solidification interface, concurrently changing the dendrites’ growth morphology. The acoustic streaming effect increases the flow velocity of the melt pool, which increases the tilt angle of the dendrites to the flow-on side and promotes the growth of secondary dendrite arms on the flow-on side. It improves the solute distribution in the melt pool and reduces elemental segregation.

Effects of field line expansion on Alfvén waves and vortices

Context. Simulations and observations of the solar atmosphere often reveal the presence of torsional Alfvén waves and vortices with sufficient power to heat the solar corona and accelerate the solar wind. Aims. We challenge the long-held view that low-frequency Alfvén waves are suppressed due to inhomogeneities and steep spatial gradients in the atmosphere. Alfvén waves and vortices in a stratified solar atmosphere are modelled with the aim of calculating and comparing their energy flux for different field line geometries. Methods. We show that the general problem of linear Alfvén wave propagation along field lines of arbitrary geometry can be reduced to a set of Klein–Gordon equations for the perturbations of the magnetic field and velocity. Solutions and corresponding energy fluxes are constructed for three cases with different expansion rates of the field lines in the lower atmosphere. Results. Expansion rates that are associated with cut-off free propagation in the lower atmosphere suppress the perturbation amplitudes and the corresponding energy flux. These include the uniform field model and the thin flux tube model. A counterexample with an intermediate field line expansion rate and non-vanishing cut-offs exhibits consistently large perturbation amplitudes and unrestricted energy flux across the entire frequency spectrum. Conclusions. Field lines with different expansion rates and geometries in the lower atmosphere can significantly alter the amplitudes of the Alfvén waves and vortices and the extent of the energy flux entering the corona.

A data-driven strategy for phase field nucleation modeling

We propose a data-driven strategy for parameter selection in phase field nucleation models using machine learning and apply it to oxide nucleation in Fe-Cr alloys. A grand potential-based phase field model, incorporating Langevin noise, is employed to simulate oxide nucleation and benchmarked against the Johnson-Mehl-Avrami-Kolmogorov model. Three independent parameters in the phase field simulations (Langevin noise strength, numerical grid discretization and critical nucleation radius) are identified as essential for accurately modeling the nucleation behavior. These parameters serve as input features for machine learning classification and regression models. The classification model categorizes nucleation behavior into three nucleation density regimes, preventing invalid nucleation attempts in simulations, while the regression model estimates the appropriate Langevin noise strength, significantly reducing the need for time-consuming trial-and-error simulations. This data-driven approach improves the efficiency of parameter selection in phase field models and provides a generalizable method for simulating nucleation-driven microstructural evolution processes in various materials.

Testing the flux tube expansion factor

Context. The properties of the solar wind measured in situ in the heliosphere are largely controlled by energy deposition in the solar corona, which is in turn closely related to the properties of the coronal magnetic field. Previous studies have shown that long-duration and large-scale magnetic structures show an inverse relation between the solar wind velocity measured in situ near 1 au and the expansion factor of the magnetic flux tubes in the solar atmosphere. Aims. The advent of the Solar Orbiter mission offers a new opportunity to analyse the relation between solar wind properties measured in situ in the inner heliosphere and the coronal magnetic field. We exploit this new data in conjunction with models of the coronal magnetic field and the solar wind to evaluate the flux expansion factor and speed relation. Methods. We use a Parker-like solar wind model, the “isopoly” model presented in previous works, to describe the motion of the solar wind plasma in the radial direction and model the tangential plasma motion due to solar rotation with the Weber and Davis equations. Both radial and tangential velocities are used to compute the plasma trajectory and streamline from Solar Orbiter location sunward to the solar ‘source surface’ at rss. We then employed a potential field source surface (PFSS) model to reconstruct the coronal magnetic field below rss to connect wind parcels mapped back to the photosphere. Results. We found a statistically weak anti-correlation between the in situ bulk velocity and the coronal expansion factor, for about 1.5 years of solar data. Classification of the data by source latitude reveals different levels of anticorrelation, which is typically higher when Solar Orbiter magnetically connects to high latitude structures than when it connects to low latitude structures. We show the existence of a fast solar wind that originates in strong magnetic field regions at low latitudes and undergoes large expansion factor. We provide evidence that such winds become supersonic during the super-radial expansion (below rss) and are theoretically governed by a positive v–f correlation. We find that faster winds exhibit, on average, a flux tube expansion at a larger radius than slower winds. Conclusions. An anticorrelation between solar wind speed and expansion factor is present for solar winds originating in high latitude structures in solar minimum activity, typically associated with coronal hole-like structures, but this cannot be generalized to lower latitude sources. We have found extended time intervals of fast solar wind associated with both large expansion factors and strong photospheric magnetic fields. Therefore, the value of the expansion factor alone cannot be used to predict the solar wind speed. Other parameters, such as the height at which the expansion gradient is the strongest, must also be taken into account.

Magnetic fields in the outskirts of PSZ2 G096.88+24.18 from a depolarization analysis of radio relics

Context. Radio relics are diffuse, non-thermal radio sources present in a number of merging galaxy clusters. They are characterized by elongated arc-like shapes and highly polarized emission (up to ∼60%) at gigahertz frequencies, and are expected to trace shock waves in the cluster outskirts induced by galaxy cluster mergers. Their polarized emission can be used to study the magnetic field properties of the host cluster. Aims. In this paper, we investigate the polarization properties of the double radio relics in PSZ2 G096.88+24.18 using the rotation measure (RM) synthesis, and try to constrain the characteristics of the magnetic field that reproduce the observed depolarization as a function of resolution (beam depolarization). Our aim is to understand the nature of the low polarization fraction that characterizes the southern relic with respect to the northern relic. Methods. We present new 1–2 GHz Karl G. Jansky Very Large Array (VLA) observations in multiple configurations. We derived the RM and polarization of the two relics by applying the RM synthesis technique, and thus solved for bandwidth depolarization in the wide observing bandwidth. To study the effect of beam depolarization, we degraded the image resolution and studied the decreasing trend of polarization fraction with increasing beam size. Finally, we performed 3D magnetic field simulations using multiple models for the magnetic field power spectrum over a wide range of scales, in order to constrain the characteristics of the cluster magnetic field that can reproduce the observed beam depolarization trend. Results. Using RM synthesis, we obtained a polarization fraction of (18.6 ± 0.3)% for the northern relic and (14.6 ± 0.1)% for the southern one. Having corrected for bandwidth depolarization, and after noticing the absence of relevant complex Faraday spectrum, we inferred that the nature of the depolarization for the southern relic is external, and possibly related to the turbulent gas distribution within the cluster, or to the complex spatial structure of the relic. The best-fit magnetic field power spectrum, which reproduces the observed depolarization trend for the southern relic, was obtained for a turbulent magnetic field model, described by a power spectrum derived from cosmological simulations, and defined within the scales of Λmin = 35 kpc and Λmax = 400 kpc. This yields an average magnetic field of the cluster within 1 Mpc3 volume of ∼2 μG.