Estimation of the 3-D geoelectric field at the Earth's surface using spherical elementary current systems

Abstract. Rapid changes in the ionospheric current system give rise to induction currents in the conducting ground that can significantly contribute to magnetic and especially electric fields at the Earth's surface. Previous studies have concentrated on the surface fields, as they are important in, for example, interpreting magnetometer measurements or in the studies of the Earth's conductivity structure. In this paper we investigate the effects of induction fields at the ionospheric altitudes for several realistic ionospheric current models (Westward Travelling Surge, Ω-band, Giant Pulsation). Our main conclusions are: 1) The secondary electric field caused by the Earth's induction is relatively small at the ionospheric altitude, at most 0.4 mV/m or a few percent of the total electric field; 2) The primary induced field due to ionospheric self-induction is locally important, ~ a few mV/m, in some "hot spots", where the ionospheric conductivity is high and the total electric field is low. However, our approximate calculation only gives an upper estimate for the primary induced electric field; 3) The secondary magnetic field caused by the Earth's induction may significantly affect the magnetic measurements of low orbiting satellites. The secondary contribution from the Earth's currents is largest in the vertical component of the magnetic field, where it may be around 50% of the field caused by ionospheric currents. Keywords. Geomagnetism and paleomagnetism (geomagnetic induction) – Ionosphere (electric fields and currents)

The geoelectric field driving geomagnetically induced currents (GIC) has complex spatial variations. It follows that different patterns of the field vectors in a given area having the same regional mean can produce very different GIC. In this study, we consider a few power grid models and calculate GIC due to a modelled geoelectric field with a regional mean magnitude of 1 V/km. Altogether 8035 snapshots of the electric field are included. We also assume two different ground conductivity models, of which the simpler one consists of two layers across the whole region, and another model has four different two-layer blocks. As a measure of GIC, we use the sum of currents at all substations of the power grid. We also consider the distribution of GIC at a single site. For a given grid, differences between the two ground conductivity models are small. When comparing different power grid models, the sum of GIC varies relatively more for a grid with a small number of substations and transmission lines. When the area and the number of substations increase, the relative difference between the smallest and largest GIC sum decreases. In all cases, assuming a spatially uniform electric field leads to a reasonable estimation of the GIC magnitudes, but it does not produce the largest GIC sum. For a single substation, there is a large variety of GIC values due to different electric field configurations.

In resonant photorefractive experiments the charge-density distribution in the conduction band is moving and an intrinsic magnetic field proportional to the velocity of the distribution is produced. In such cases the free electron plasma can interact with an externally applied magnetic field. On top of that materials exposed to a magnetic field exhibit Faraday rotation (Voigt birefringence) of the state of polarization of the optical bean.*. The purpose of the present summary is to describe the effect of both phenomena in a standard wave mixing setup. The photorefractive band transport model in the case v here magnetic fields are present is only modified via a term in the current density equation and reads (1) where e is the numeric elementary charge, µ the free electron mobility, n the number density' of free electrons, E the total electric field which may consist of an externally applied term and the norihnear generated spacecharge field, D the diffusion constant, and B the magnetic field term which may consist of an external part and an internal part (stemming from electronic motion). The remaining equations describing the photorefractive effect are the conventional ones. Now, by inferring the small modulation approximation, a perturbational relation for the physical quantities and, by Fourie- transforming in space-time, a steady-state response function for the space-charge electric field are found. ThL photorefractive frequency response function, expressed as a function of the conventional photorefractive parameters, is given by where the function P is given by (3) The actual definition of the parameters in Eqs. (2) and (3) can be found in Ref. [1]. Equation (2) is taken in the case where an externally magnetic field is applied perpendicular to the grating wave vector K. If it is assumed that the constituents of the incident intensity are monochromatic plane waves, that may be frequency shifted a small amount, the Fourier transformation of the perturbed intensity is a sum of δ-functions peaking at k = ±K and at ω = ±Ω. The space-charge field is now given by (4) A typical response function for GaAs:Cr is shown in Fig. 1. The characteristic parameters used in the calculation are taken from Refs. [2, 3, 4], The externally applied electric field is Eθ = 10 V/m, the applied magnetic field is Bθ = 2 T, the angle between the applied electric field and the grating wave vector is 20°, and the angle between the applied magnetic field and the z-axis is 45°. Recently, experiments on photorefr?ctive wave mixing with Faraday rotation in a diluted magnetic semiconductor of Cd] MnTe have been demonstrated. The gain showed an oscillatory behaviour as a function of the' magnetic field, and it was demonstrated that the magnetic field controls the direction and magnitude of the energy transfer. In what follows the coupled wave equations including uie effect of optical activity, the Faraday effect, and the effect of linear absorption will be given. Mcreover, the magnetic effects in the space-charge field given above will be taken into account. Numeric solutions to the coupled differential equations will be given. The equations are given in the following form (5) (6) (7) (8) where α is the absorption constant, Γθ the birefringence coupling constant, p the rotatory power, V the Verdet constant, <5 the angle between the z-axis and the magnetic field vector, and Γ (K,Ω) the photorefractive coupling parameter. A(. and are the polarization components perpendicular and^ parallel to the plane of incidence, respectively. In Fig. 2 a typical numerical solution to the above equations is given for photorefractive GaAs:Cr. The applied magnetic field is Bθ = 2 T, The electric field is Eθ - 106 V/m, the grating spacing is A - 34 µm, and the frequency shift is Ω = 94 rad/s. In such a material the optical activity is zero so the oscillations are due to the Faraday effect. Figure 1. The space-charge response function for GaAs:Cr. Figure 2. Intensity of the probe beam in GaAs:Cr as a function of distance in the crystal for a 10 mm crystal.

Transformers in New Zealand's South Island electrical transmission network have been impacted by geomagnetically induced currents (GIC) during geomagnetic storms. We explore the impact of GIC on this network by developing a thin‐sheet conductance (TSC) model for the region, a geoelectric field model, and a GIC network model. (The TSC is composed of a thin‐sheet conductance map with underlying layered resistivity structure.) Using modeling approaches that have been successfully used in the United Kingdom and Ireland, we applied a thin‐sheet model to calculate the electric field as a function of magnetic field and ground conductance. We developed a TSC model based on magnetotelluric surveys, geology, and bathymetry, modified to account for offshore sediments. Using this representation, the thin sheet model gave good agreement with measured impedance vectors. Driven by a spatially uniform magnetic field variation, the thin‐sheet model results in electric fields dominated by the ocean‐land boundary with effects due to the deep ocean and steep terrain. There is a strong tendency for the electric field to align northwest‐southeast, irrespective of the direction of the magnetic field. Applying this electric field to a GIC network model, we show that modeled GIC are dominated by northwest‐southeast transmission lines rather than east‐west lines usually assumed to dominate.

In the marine environment, active and passive electromagnetic (EM) measurements are used to derive information about the conductivity structure beneath the seafloor. While the conductivity is mostly determined by the conductive seawater contained in the pore space or fractures, anomalies may occur in the presence of more resistive (e.g. hydrocarbons, free gas, gas hydrates, freshened water) or conductive materials (e.g. massive sulfides, brines). For the correct interpretation of EM data it is important to know the measurement geometry, including the orientations of receivers. From an experimental standpoint, this can be challenging because stations are often deployed free falling, thus, ending up in arbitrary orientations on the seafloor. The orientations are frequently derived from electronic compass measurements or magnetometers which record all components of the magnetic field. However, these measurements may be distorted by magnetic parts on stations (e.g. batteries), biased by local inhomogeneities in the local field or difficult to perform if no reliable reference data from a nearby observatory is available.A possible remedy for such problems may come from space physics. Given a grid of stationary magnetometer stations, surrounding the area of interest but at relatively large distances, the method of spherical elementary current systems (SECS) (Amm & Viljanen, 1999, Earth, Plants and Space) can be used to reconstruct equivalent ionospheric currents and their resulting time variations of the magnetic field at any point within the grid. The method is especially suitable to be applied at northern latitudes, where fairly dense magnetometer networks such as IMAGE and CARISMA exist, and where the magnitude of geomagnetic disturbances from ionospheric currents is significant.We have successfully applied the method to three marine EM data sets, one offshore Iceland, one in the arctic section of the North Atlantic (Loki's Castle) and one off the Canadian coast (Prince Edward Island). The SECS method qualitatively reproduces the magnetic variation as observed by the seafloor stations. Here we investigate the results from the above mentioned EM data sets, and discuss the applicability, accuracy and constraints of the SECS method for EM data calibrations. Furthermore, we illuminate the possibility for using SECS as an interpolation tool for other applications at remote offshore locations, such as measurement while drilling (MWD) operations.

This paper describes the development of a system for simulation of the geomagnetically induced currents (GIC) produced in a power system. Magnetic field data or a specified electrojet current are used with an Earth conductivity model to calculate the electric fields at the Earth's surface. These electric field values are used as input to a power system model to give the geomagnetically induced currents throughout the network. Practical calculations can be made by superposition of results calculated separately for northward and eastward electric fields. Superposition can be used with magnetic data from a single observatory for uniform fields, or with data from two observatories assuming a linear variation of the magnetic fields between the two observatory locations. Combining these calculations provides an efficient method for simulating the geomagnetically induced currents produced during any disturbance for which geomagnetic data are available.

The proliferation of ultra‐high voltage direct current (UHVDC) transmission technology has heightened regulatory oversight of electromagnetic fields around buildings. In conventional direct current (DC) line design, the presence of residential areas was typically not taken into account. GB 39220‐2020 now requires environmental assessments for residential balconies and platforms. The impact of nearby residential buildings has to be taken into account during the design of DC lines, necessitating the calculation of the three‐dimensional total electric field. This study introduces a parallel charge simulation method‐flux tracing method to quickly calculate the 3D total electric field near buildings in proximity to UHVDC lines, which is crucial for engineering design and environmental assessment. The method significantly speeds up calculations, with a 4–15 times faster ratio using 16–32 threads. It also analyses how UHVDC line dimensions, distances to buildings and parameters of residential buildings affect the maximum electric field on platforms. By using regression analysis, the formula for predicting the maximum electric field is obtained, and the traditional calculation time is improved. This advancement provides efficient predictions and detailed 3D electric field distributions for engineering and environmental assessments.

High temporal and high spatial resolution geoelectric field models of two Mäntsälä, Finnish pipeline geomagnetically induced current (GIC) intervals that occurred within the 7–8 September 2017 geomagnetic storm have been made. The geomagnetic measurements with 10 s sampling rate of 28 IMAGE ground magnetometers distributed over the north Europe (from 52.07° to 69.76° latitude) are the bases for the study. A GeoElectric Dynamic Mapping (GEDMap) code was developed for this task. GEDMap considers 4 different methods of interpolation and allows a grid of 0.05° (lat.) × 0.2° (lon.) spatial scale resolution. The geoelectric field dynamic mapping output gives both spatial and temporal variations of the magnitude and direction of fields. The GEDMap results show very rapid and strong variability of geoelectric field and the extremely localized peak enhancements. The magnitude of geoelectric fields over Mäntsälä at the time of the two GIC peaks were 279.7 and 336.9 mV/km. The comparison of the GIC measurements in Mäntsälä and our modeling results show very good agreement with a correlation coefficient higher than 0.8. It is found that the auroral electrojet geoelectric field has very rapid changes in both magnitude and orientation causing the GICs. It is also shown that the electrojet is not simply oriented in the east‐west direction. It is possible that even higher time resolution base magnetometer data of 1 s will yield even more structure, so this will be our next effort.

During geomagnetic storms the electrical power grid is vulnerable to geomagnetically induced currents (GICs) caused by sharp changes in magnetic and geoelectric fields. In the UK the measurement of GIC in the power grid is extremely limited with most GIC estimates coming from a model of the high voltage grid. The US has collected geomagnetic disturbance (GMD) data for 18 geomagnetic storms. Network theory is routinely used to estimate the resilience of the physical power grid, and its robustness to the removal of nodes, when faced with threats ranging from natural hazards to cyber-attacks but is currently not applied to GIC. By applying network theory to both the modelled UK dataset and measured US dataset, we can utilize known parameters to test for vulnerabilities to space weather in the power grid across varying spatial and temporal scales. The network is formed using methods of association between the GIC data at each transformer. The monitors are the nodes of the network and the links are defined as when the wavelet cross-correlation of the GIC is sufficiently high (1). The wavelet transform is used to localise the GIC response to the storm across time scales. Whilst previous network science studies have focused on the physical topology of the power grid, our method focuses on the dynamical response of the grid to GIC. Despite the difference in latitude and local time we see many similarities between the modelled UK and measured US GIC data, particularly during the sudden commencement. Initial results show the same nodes repeatedly appearing as the most highly connected to the network across multiple events. These nodes could be key to providing resilience and/or prediction of forthcoming disturbance of the power grid in the event of a large geomagnetic storm.(1) Orr, L., Chapman, S. C., Beggan, C. D. (2021). Space Weather, 19, e2021SW002772.

We have derived comprehensive statistics of geomagnetic activity for assessing the occurrence of geomagnetically induced currents (GIC) in the Norwegian high-voltage power grid. The statistical study is based on geomagnetic recordings in 1994–2011 from which the geoelectric field can be modelled and applied to a DC description of the power grid to estimate GIC. The largest GIC up to a few 100 A in the Norwegian grid occur most likely in its southern parts. This follows primarily from the structure of the grid favouring large GIC in the south. The magnetic field has its most rapid variations on the average in the north, but during extreme geomagnetic storms they reach comparable values in the south too. The ground conductivity has also smaller values in the south, which further increases the electric field there. Additionally to results in 1994–2011, we performed a preliminary estimation of a once per 100 year event for geoelectric field by extrapolating the statistics. We found that the largest geoelectric field value would be twice the maximum in 1994–2011. Such value was actually reached on 13–14 July 1982.

Previous studies have demonstrated a close relationship between the time derivative of the horizontal geomagnetic field vector (dH /dt ) and geomagnetically induced currents (GIC) at a nearby location in a power grid. Similarly, a high correlation exists between GIC and the local horizontal geoelectric field (E ), typically modelled from a measured magnetic field.Considering GIC forecasting, it is not feasible to assume that detailed prediction of time series will be possible. Instead, other measures summarising the activity level over a given period are preferable. In this paper, we consider the 30-min maximum of dH /dt or E as a local activity indicator (|dH /dt |30 or |E |30 ). Concerning GIC, we use the sum of currents through the neutral leads at substations and apply its 30-min maximum as a regional activity measure (GIC30 ).We show that |dH /dt |30 at a single point yields a proxy for GIC activity in a larger region. A practical consequence is that if |dH /dt |30 can be predicted at some point then it is also possible to assess the expected GIC level in the surrounding area. As is also demonstrated, |E |30 and GIC30 depend linearly on |dH /dt |30 , so there is no saturation with increasing geomagnetic activity contrary to often used activity indices.

Aggregate stability is a key factor in the evaluation of soil structure and erosion resistance, which is largely influenced by soil electric field and particle interaction. However, there are few studies on how different organic and inorganic soil conditioners change the surface electric field and interaction force of reclaimed soil to improve the aggregate stability. Therefore, a five-year field experiment was conducted to quantitatively study the effects of FeSO4 (TM), organic fertilizer (TO), fly ash (TF), maturing agent + organic fertilizer (TMO) and fly ash + organic fertilizer (TFO), compared with control (CK) treatment, on the reclaimed soil internal force and the aggregate crushing strength. The results showed that the reclaimed soil surface potential and electric field intensity increased after 5 years of application of organic and inorganic soil conditioners. Under the same electrolyte concentration and electric field conditions, the crushing strength of aggregates (<5 μm) treated with TFO, TMO, TO, TF and TM decreased by 43.70%, 35.51%, 25.97%, 8.28% and 5.49%, respectively, compared with the control treatment, and the combination of organic and inorganic treatment (TFO and TMO) had a better effect on improving the aggregate crushing resistance. With the application of soil conditioners, the reclaimed soil DLVO force and net resultant force gradually decreased, and the order of magnitude was TFO < TMO < TO < TF < TM < CK, indicating that the application of organic and inorganic soil conditioners enhanced the van der Waals attractive force and net attractive force between reclaimed soil particles, and reduced the net repulsive force between particles. The theoretical calculation results of the reclaimed soil internal force well explain the experimental results of aggregate stability against crushing, and the relationship between aggregate crushing strength and net resultant force is exponential (p < 0.01). Generally speaking, the soil conditioners increase the net attractive force between particles, reduce the possibility of violent crushing of aggregates due to the increase of electric field intensity and improve the aggregate structural stability, among which the combined application of organic and inorganic soil conditioners has a better improvement effect. The results of this study will lay a theoretical foundation for clarifying the improvement of different soil conditioners on the reclaimed soil structural stability and erosion resistance.

Sweden has experienced many geomagnetically induced current (GIC) events in the past, which is obviously due to the high‐latitude location of the country. The largest GIC, almost 300 A, was measured in southern Sweden in the earthing lead of a 400 kV transformer neutral during the magnetic storm on 6 April 2000. On 30 October 2003, the city of Malmö at the southern coast suffered from a power blackout caused by GIC, leaving 50,000 customers without electricity for about 20–50 min. We have developed a model that enables calculation of GIC in the southern Swedish 400 kV power grid. This work constitutes the first modeling effort of GIC in Sweden. The model is divided into two parts. The electric field is first derived using a ground conductivity model and geomagnetic recordings from nearby stations. The conductivity model is determined from a least squares fit between measured and calculated GIC. GIC are calculated using a power grid model consisting of the topology of the system and of the transformer, transmission line, and station earthing resistances as well as of the coordinates of the stations. To validate the model, we have compared measured and calculated GIC from one site. In total, 24 events in 1998 to 2000 were used. In general the agreement is satisfactory as the correct GIC order of magnitude is obtained by the model, which is usually enough for engineering applications.

Microwave breast hyperthermia is a noninvasive treatment method for breast cancer that utilizes microwave energy (ME) sources to raise tissue temperatures above 42 , inducing tumor cell necrosis. The efficiency of ME deposition depends on the electric field magnitude and tissue conductivity, with antenna phase and amplitude adjustments used to maximize the electric field magnitude within tumors. Achieving precise ME focusing in the complex and heterogeneous breast tissue is challenging and can lead to unwanted hot spots in normal tissue. This study presents a novel method for optimizing ME focusing on the center of target tumors, using a simplified calculation of antenna phases, heuristic optimization for antenna amplitudes, and realistic breast phantoms for performance evaluation. In this work, we propose an approach to optimize the microwave hyperthermia system, employing phase and amplitude modulation techniques to concentrate the electric field at the center of a malignant tumor within a breast medium. The approach uses line sources arranged in a circular pattern around realistic breast models. The method begins by determining the phase, followed by adjusting the amplitudes of each source in order to maximize the total electric field at the tumor's center. The goal is to maximize the electric field at the tumor center while minimizing the optimization cost andcomplexity. Simulations are performed at 4 GHz frequency using two different types of digital breast phantoms (fatty and dense breasts) as test beds. The algorithm is tested by using three quantities; that is, the electric field distribution, the power density distribution, and the temperature distribution inside the whole breast region. The electric field and power density are calculated using an in-house method of moments (MoM) algorithm, while the temperature distributions are obtained with computer simulation technology (CST) software. To further evaluate the method with quantitative measures of success, thermal indices are calculated for each phantom andmethod. The specific absorbtion rate (SAR) results and corresponding temperature distributions for each breast type and optimization demonstrate that effective focusing is achieved in both cases. However, the combined phase-amplitude optimization provides more precise focusing by eliminating hot spots. Among thermal indices, the TC75 and T90 values obtained from the phase-amplitude combined optimization for both breast types outperform the results found in the literature. The T50 values obtained using the combined optimization are above 42 . This study presents an optimization method for focusing ME within breast tissue, performed in two steps: first phase optimization, followed by amplitude optimization. The electric field calculations are performed using both the MoM and Finite Difference Time Domain methods. The technique is numerically tested on two realistic breast models, with thermal indices calculated for each phantom and optimization process. Results show T90 values exceeding 40 and T50 values above 42 . While the study employs a 2D applicator, it provides a strong foundation for future development in 3Dapplications.

A technique for estimating a plasma drift velocity distribution in the ionosphere is presented. This technique is based on a framework for representing a global vector field on a sphere by using a set of localized basis functions which is newly derived as a variant of the spherical elementary current system (SECS). A vector field on a sphere can be divided into its divergence-free (DF) component and curl-free (CF) component. The DF and CF components can then be represented by weighted sums of the DF and CF vector-valued basis functions, respectively. While the SECS basis functions have a singular point, the new basis functions do not diverge over a sphere. This property of the new basis function allows us to achieve robust prediction of the drift velocity at any point in the ionosphere. Assuming that the ionospheric plasma drift velocity has no divergence, its distribution can be represented by a weighted sum of the DF basis functions. The proposed technique estimates the ionospheric plasma drift velocity distribution from the SuperDARN data by using the DF basis functions. Since there are some wide gaps in the spatial coverage of the SuperDARN, an empirical convection model is combined with the framework based on the new basis functions. It is demonstrated that the proposed technique is useful for the estimation and modeling of the ionospheric plasma velocity distribution.