Time Derivatives Of Fields Research Articles

Space Weather Disturbances in Non‐Stormy Times: Occurrence of dB/dt Spikes During Three Solar Cycles

AbstractSpatio‐temporal variations of ionospheric currents cause rapid magnetic field variations at ground level and Geomagnetically Induced Currents (GICs) that can be harmful for human infrastructure. The risk for large excursions in the magnetic field time derivative, “dB/dt spikes”, is known to be high during geomagnetic storms and substorms. However, less is known about the occurrence of spikes during non‐stormy times. We use data from ground‐based globally covering magnetometers (SuperMAG database) from the years 1985–2021. We investigate the spike occurrence (|dB/dt| > 100 nT/min) as a function of magnetic local time (MLT), magnetic latitude (Mlat), and the solar cycle phases during non‐stormy times (−15 nT ≤ SYM‐H < 0). We sort our data into substorm (AL < 200 nT) intervals (“SUB”) and less active intervals between consecutive substorms (“nonSUB”). We find that spikes commonly occur in both SUBs and nonSUBs during non‐stormy times (3–23 spikes/day), covering 18–12 MLT and 65°–80° Mlat. This also implies a risk for infrastructure damage during non‐stormy times, especially when several spikes occur nearby in space and time, possibly causing infrastructure weathering. We find that spikes are more common in the declining phase of the solar cycle, and that the occurrence of SUB spikes propagates from one midnight to one morning hotspot with ∼10 min in MLT for each minute in universal time (UTC). Finally, we discuss causes for the spikes in terms of spatio‐temporal variations of ionospheric currents.

A new radiation reaction approximation for particle dynamics in the strong field regime

Context. Following particle trajectories in the intense electromagnetic field of a neutron star is prohibited by the large ratio between the cyclotron frequency ωB and the stellar rotation frequency Ω. No fully kinetic simulations on a macroscopic scale and with realistic field strengths have been performed so far due to the huge computational cost implied by this enormous scale of separation. Aims. In this paper, we derive new expressions for the particle velocity subject to strong radiation reaction that are intended to be more accurate than the current state-of-the-art expression in the radiation reaction limit regime, the so-called Aristotelian regime. Methods. We shortened the timescale hierarchy by solving the particle equation of motion in the radiation reaction regime, where the Lorentz force is always and immediately balanced by the radiative drag, and including a friction not necessarily opposite to the velocity vector, as derived in the Landau-Lifshitz approximation. Results. Starting from the reduced Landau-Lifshitz equation (i.e., neglecting the field time derivatives), we found expressions for the velocity depending only on the local electromagnetic field configuration and on a new parameter related to the field strength that controls the strength of the radiative damping. As an example, we imposed a constant Lorentz factor γ during the particle motion. We found that for ultra-relativistic velocities satisfying γ ≳ 10, the difference between strong radiation reaction and the radiation reaction limit becomes negligible. Conclusions. The new velocity expressions produce results similar in accuracy to the radiation reaction limit approximation. We therefore do not expect this new method to improve the accuracy of neutron star magnetosphere simulations. The radiation reaction limit is a simple but accurate, robust, and efficient way to follow ultra-relativistic particles in a strong electromagnetic field.

A Locally-Implicit Discontinuous Galerkin Time-Domain Method to Simulate Metasurfaces Using Generalized Sheet Transition Conditions

The generalized sheet transition conditions (GSTCs) are incorporated into a discontinuous Galerkin time-domain (DGTD) method to efficiently simulate metasurfaces. The numerical flux for GSTCs is derived for the first time using the Rankine–Hugoniot jump conditions. This numerical flux includes the time derivatives of fields, and therefore, explicit time integration schemes, which are traditionally used with DGTD, do not yield a stable time marching method. To alleviate this bottleneck, a new time marching scheme, which solves a local matrix system for the unknowns of the elements touching the same GSTC face, is developed. This locally implicit method maintains its high-parallel efficiency just like the traditional explicit DGTD schemes. Numerical results, which validate the accuracy of the proposed method against analytical solutions and demonstrate its applicability to the simulation of curved and space/time-varying metasurfaces, are presented.

Gauging the higher derivative field theories in Ostrogradsky formalism

The gaugings of the higher derivative fields theory are analyzed in Ostrogradsky formalism with the aid of BRST charge. Treating the terms of higher-order time derivatives of fields as independent dynamic variables, we are able to convert the higher derivative Lagrangian into the more suitable Ostrogradski Hamiltonian formalism. Using the knowledge of local BRST-cohomology, it is possible for us to work out the deformed functional equations of the BRST charge as well as the corresponding BRST-invariant Hamiltonian. Moreover, by inserting these deformed quantities into the original system and utilizing the canonical equations of motion of phase variables, we gain the usual local gauge symmetries of the resulting interacting theory directly through the standard Legendre transformations.

Nowcasting and Validating Earth's Electric Field Response to Extreme Space Weather Events Using Magnetotelluric Data: Application to the September 2017 Geomagnetic Storm and Comparison to Observed and Modeled Fields in Scotland

AbstractIn the United Kingdom, geomagnetically induced currents (GICs) are calculated from thin‐sheet electrical conductivity models. In the absence of conductivity models, time derivatives of magnetic fields are sometimes used as proxies for GIC‐related electric fields. An alternative approach, favored in the United States, is to calculate storm time electric fields from time‐independent impedance tensors computed from an array of magnetotelluric (MT) sites and storm time magnetic fields recorded at geomagnetic observatories or assumed from line current models. A paucity of direct measurements of storm time electric fields has restricted validation of these different techniques for nowcasting electric fields and GICs. Here, we present unique storm time electric field data from seven MT sites in Scotland that recorded before, during, and after the September 2017 magnetic storm. By Fourier transforming electric field spectra computed using different techniques back to the time domain, we are able to make direct comparisons with these measured storm time electric field time series. This enables us to test the validity of different approaches to nowcasting electric fields. Our preferred technique involves frequency domain multiplication of magnetic field spectra from a regional site with a local impedance tensor that has been corrected for horizontal magnetic field gradients present between the local site and the regional site using perturbation tensors derived from geomagnetic depth sounding (GDS). Scatter plots of scaling factors between measured and nowcasted electric fields demonstrate the importance of coupling between the polarization of the storm time magnetic source field and Earth's direction‐dependent deep electrical conductivity structure.

Improved single-particle phase-space distributions for viscous fluid dynamic models of relativistic heavy ion collisions

Comparison of hydrodynamic calculations with experimental data inevitably requires a model for converting the fluid to particles. For shear viscous fluids, several different conversion models have been proposed, such as the quadratic Grad corrections, the Strickland-Romatschke (SR) ansatz, self-consistent shear corrections from linearized kinetic theory, or corrections from the relaxation time approach. We show, via comparison to nonlinear $2\ensuremath{\rightarrow}2$ kinetic theory evolution, that even for low specific shear viscosities $0.03\ensuremath{\lesssim}\ensuremath{\eta}/s\ensuremath{\lesssim}0.2$, these four models have varying accuracy and still significant errors in reproducing particle phase space densities from hydrodynamic fields. Moreover, we demonstrate that the overall reconstruction error of additive shear viscous $f={f}_{\mathrm{eq}}+\ensuremath{\delta}f$ models dramatically reduces by up to one order of magnitude, if one ensures through exponentiation that $f$ is always positive. We also illustrate how even more accurate viscous $\ensuremath{\delta}f$ models can be constructed by incorporating the first time derivatives of hydrodynamic fields to account for the past evolution of the system. Such time derivatives are readily available in hydrodynamic simulations, though usually not included in the output. Though our comparisons are limited to a one-component massless system undergoing a $(0+1)$-dimensional $(0+1\mathrm{D})$ longitudinal boost-invariant expansion, we expect that the improved models will be useful in $2+1\mathrm{D}$ and $3+1\mathrm{D}$ hydrodynamic studies as well.

Modeling Compact Intracloud Discharge (CID) as a Streamer Burst

Narrow Bipolar Pulses are generated by bursts of electrical activity in the cloud and these are referred to as Compact Intracloud Discharges (CID) or Narrow Bipolar Events in the current literature. These discharges usually occur in isolation without much electrical activity before or after the event, but sometimes they are observed to initiate lightning flashes. In this paper, we have studied the features of CIDs assuming that they consist of streamer bursts without any conducting channels. A typical CID may contain about 109 streamer heads during the time of its maximum growth. A CID consists of a current front of several nanosecond duration that travels forward with the speed of the streamers. The amplitude of this current front increases initially during the streamer growth and decays subsequently as the streamer burst continues to propagate. Depending on the conductivity of the streamer channels, there could be a low-level current flow behind this current front which transports negative charge towards the streamer origin. The features of the current associated with the CID are very different from those of the radiation field that it generates. The duration of the radiation field of a CID is about 10–20 μs, whereas the duration of the propagating current pulse associated with the CID is no more than a few nanoseconds in duration. The peak current of a CID is the result of a multitude of small currents associated with a large number of streamers and, if all the forward moving streamer heads are located on a single horizontal plane, the cumulative current that radiates at its peak value could be about 108 A. On the other hand, the current associated with an individual streamer is no more than a few hundreds of mA. However, if the location of the forward moving streamer heads are spread in a vertical direction, the peak current can be reduced considerably. Moreover, this large current is spread over an area of several tens to several hundreds of square meters. The study shows that the streamer model of the CID could explain the fine structure of the radiation fields present both in the electric field and electric field time derivative.

The impact of coronal mass ejection on the horizontal geomagnetic fields and the induced geoelectric fields

This work investigates the influence of coronal mass ejection (CME) on the time derivatives of horizontal geomagnetic and geoelectric fields, proxy parameters for identifying GICs. 16 events were identified for the year 2003 from the CORONAS-PHOTON spacecraft. Five of the events (May 29, June 9, October 28, October 29, and November 4) were extensively discussed over four magnetic observatories, were analyzed using the time derivatives of the horizontal geomagnetic (dH/dt) and geoelectric (EH) fields obtained from data of the INTERMAGNET network. It was observed that energy distributions of the wavelet power spectrum of the horizontal geoelectric field are noticed at the nighttime on both 29 May and 9 June 2003 across the stations. Daytime and nighttime intensification of energy distribution of the wavelet power spectrum of the horizontal geoelectric field are observed on both 28 and 29 October 2003 due to strong westward electrojet. The 4 November 2003 event depicts daytime amplification of energy distributions of the wavelet power spectrum across the stations. The highest correlation magnitude is obtained in the event of 4 November 2003 between dH/dt and EH relationships during the intense solar flare of class X 17.4. We observed that the correlation magnitude between dH/dt and EH increases with increase in CME activity. We concluded that the response of the surface impedance model for different stations plays a key role in determining the surface electric field strength, due to large electric field changes at different stations.

Consistency relations for large-scale structures: Applications for the integrated Sachs-Wolfe effect and the kinematic Sunyaev-Zeldovich effect

Consistency relations of large-scale structures provide exact nonperturbative results for cross-correlations of cosmic fields in the squeezed limit. They only depend on the equivalence principle and the assumption of Gaussian initial conditions, and remain nonzero at equal times for cross-correlations of density fields with velocity or momentum fields, or with the time derivative of density fields. We show how to apply these relations to observational probes that involve the integrated Sachs-Wolfe effect or the kinematic Sunyaev-Zeldovich effect. In the squeezed limit, this allows us to express the three-point cross-correlations, or bispectra, of two galaxy or matter density fields, or weak lensing convergence fields, with the secondary cosmic microwave background distortion in terms of products of a linear and a nonlinear power spectrum. In particular, we find that cross-correlations with the integrated Sachs-Wolfe effect show a specific angular dependence. These results could be used to test the equivalence principle and the primordial Gaussianity, or to check the modeling of large-scale structures.

Non-invasive monitoring system of synchronous generator using external field

This paper presents an equipment dedicated to the monitoring of synchronous generator through the measurements of the external magnetic field time derivative. The equipment uses a methodology which can be an additional technique to those applied currently on the faults detection, mainly when monitoring the incipient faults type. Based on frequency spectra (magnetic signatures), the faults detection is performed by the analysis of the components evolution. As it is not possible to test the ability of the equipment in detecting several types of faults of generators faults in a power plant, a testbench was constructed for this purpose. Faults inserted deliberately in the generators of the testbench are detected by the equipment, showing its capacity. This equipment has been also installed in power plants.

Forecasting auroras from regional and global magnetic field measurements

Abstract. We use the connection between auroral sightings and rapid geomagnetic field variations in a concept for a Regional Auroral Forecast (RAF) service. The service is based on statistical relationships between near-real-time alerts issued by the NOAA Space Weather Prediction Center and magnetic time derivative (dB∕dt) values measured by five MIRACLE magnetometer stations located in Finland at auroral and sub-auroral latitudes. Our database contains NOAA alerts and dB∕dt observations from the years 2002–2012. These data are used to create a set of conditional probabilities, which tell the service user when the probability of seeing auroras exceeds the average conditions in Fennoscandia during the coming 0–12 h. Favourable conditions for auroral displays are associated with ground magnetic field time derivative values (dB∕dt) exceeding certain latitude-dependent threshold values. Our statistical analyses reveal that the probabilities of recording dB∕dt exceeding the thresholds stay below 50 % after NOAA alerts on X-ray bursts or on energetic particle flux enhancements. Therefore, those alerts are not very useful for auroral forecasts if we want to keep the number of false alarms low. However, NOAA alerts on global geomagnetic storms (characterized with Kp values > 4) enable probability estimates of > 50 % with lead times of 3–12 h. RAF forecasts thus rely heavily on the well-known fact that bright auroras appear during geomagnetic storms. The additional new piece of information which RAF brings to the previous picture is the knowledge on typical storm durations at different latitudes. For example, the service users south of the Arctic Circle will learn that after a NOAA ALTK06 issuance in night, auroral spotting should be done within 12 h after the alert, while at higher latitudes conditions can remain favourable during the next night.

Propagation Effects on the Electric Field Time Derivatives Generated by the Stepped Leaders and Return Strokes in CG Flashes

In this paper, the effects of propagation over finitely conducting ground on the characteristics of the electric-field time-derivative (dE/dt) pulses, which are generated by the stepped leaders and return strokes in cloud-to-ground (CG) flashes, are investigated. First, a finite-difference time-domain (FDTD) method in two-dimensional (2-D) cylindrical coordinates is proposed. And then, the electric fields produced by different lightning processes are calculated. The current waveform parameters of the stepped leaders are derived based on the MTLE model by trial and error, given the analytical waveform and the measured data at the International Center for Lightning Research and Testing (ICLRT) in Florida, USA. In conclusion, the results demonstrate that the characteristics of the dE/dt pulses, including the peak amplitude, the half-width time, and the 10%-90% rise time, change significantly in propagating over finitely conducting ground. Furthermore, the effects of the parameters (i.e., 10%-90% rise time and half-width time) of the lightning current waveforms on the characteristics of dE/dt pulses are also discussed. The results indicate that the 10%-90% rise time, as compared with the half-width time, has a more significant influence on the characteristics of the dE/dt pulses. Lastly, average energy spectra densities of the simulated dE/dt pulses are calculated, and for comparison purposes, spectral analysis of the data obtained by the very high frequency broadband interferometer are also presented. The theoretical results are consistent with those available experimental data.

Mapping geomagnetic secular variation at the core-mantle boundary

SUMMARY Data from recent satellite missions have vastly increased the resolution of models of the geomagnetic field, and itsfirst and second time derivatives – secular variation (SV) and secular acceleration (SA). The spectra of both SV and SA are ‘blue’ at the core–mantle boundary, both well-fit by functions proportional to l(l + 1) where l is the spherical harmonic degree. The ratio of the two spectra defines a timescale for geomagnetic variations of approximately 10 yrs for all resolvable harmonic degrees. The blue spectra should prevent meaningful maps of the SV being generated; nevertheless, the coherence of the maps up to harmonic degree 13 suggests that it is possible to obtain useful insight from their examination. Low SV is confirmed under the Pacific, but also revealed under the North Atlantic and Antarctica. These features are more readily explained in terms of dynamo control through thermal core–mantle coupling than by electromagnetic screening. Comparison with maps from measurements prior to the recent satellites, using the ‘Comprehensive Model’, suggests that models back to at least 1970 are sufficiently good to enable direct comparison of the SV.

3D time-domain simulation of electromagnetic diffusion phenomena: A finite-element electric-field approach

We present a finite-element time-domain (FETD) approach for the simulation of 3D electromagnetic (EM) diffusion phenomena. The finite-element algorithm efficiently simulates transient electric fields and the time derivatives of magnetic fields in general anisotropic earth media excited by multiple arbitrarily configured electric dipoles with various signal waveforms. To compute transient electromagnetic fields, the electric field diffusion equation is transformed into a system of differential equations via Galerkin’s method with homogeneous Dirichlet boundary conditions. To ensure numerical stability and an efficient time step, the system of the differential equations is discretized in time using an implicit backward Euler scheme. The resultant FETD matrix-vector equation is solved using a sparse direct solver along with a fill-in reduced ordering technique. When advancing the solution in time, the FETD algorithm adjusts the time step by examining whether or not the current step size can be doubled without unacceptably affecting the accuracy of the solution. To simulate a step-off source waveform, the 3D FETD algorithm also incorporates a 3D finite-element direct current (FEDC) algorithm that solves Poisson’s equation using a secondary potential method for a general anisotropic earth model. Examples of controlled-source FETD simulations are compared with analytic and/or 3D finite-difference time-domain solutions and are used to confirm the accuracy and efficiency of the 3D FETD algorithm.

Characterization of the plasma current quench during disruptions in the National Spherical Torus Experiment

A detailed analysis of the plasma current quench in the National Spherical Torus Experiment (Ono et al 2000 Nucl. Fusion 40 557) is presented. The fastest current quenches are fit better by a linear waveform than an exponential one. Area-normalized current quench times down to 0.4 ms m−2 have been observed, compared with the minimum of the 1.7 ms m−2 recommendation based on conventional aspect ratio tokamaks; as noted in previous ITPA studies, the difference can be explained by the reduced self-inductance at low aspect ratio and high elongation. The maximum instantaneous dIP/dt is often many times larger than the mean quench rate, and the plasma current before the disruption is often substantially less than the flat-top value. The poloidal field time derivative during the disruption, which is directly responsible for driving eddy currents, has been recorded at various locations around the vessel. The IP quench rate, plasma motion and magnetic geometry all play important roles in determining the rate of poloidal field change.

Propagation effects on the electric field time derivatives generated by return strokes in lightning flashes

The effects of propagation over finitely conducting ground on the features of radiation component of the electric field time derivatives are investigated. The results show that the peak, the half-width and the risetime of the electric field time derivative change significantly in propagating over finitely conducting ground. Furthermore, any correlation that may exist between various parameters could also change significantly due to propagation effects. Consequently, in return stroke model validations using experimentally measured fields, remote sensing of return stroke current time derivatives using measured electric field time derivatives and in the calculation of induced voltages generated by lightning flashes in electrical installations the distortions caused by propagation effects on the electric field time derivatives cannot be neglected.

A parallel finite‐difference approach for 3D transient electromagnetic modeling with galvanic sources

A parallel finite‐difference algorithm for the solution of diffusive, three‐dimensional (3D) transient electromagnetic field simulations is presented. The purpose of the scheme is the simulation of both electric fields and the time derivative of magnetic fields generated by galvanic sources (grounded wires) over arbitrarily complicated distributions of conductivity and magnetic permeability. Using a staggered grid and a modified DuFort‐Frankel method, the scheme steps Maxwell's equations in time. Electric field initialization is done by a conjugate‐gradient solution of a 3D Poisson problem, as is common in 3D resistivity modeling. Instead of calculating the initial magnetic field directly, its time derivative and curl are employed in order to advance the electric field in time. A divergence‐free condition is enforced for both the magnetic‐field time derivative and the total conduction‐current density, providing accurate results at late times. In order to simulate large realistic earth models, the algorithm has been designed to run on parallel computer platforms. The upward continuation boundary condition for a stable solution in the infinitely resistive air layer involves a two‐dimensional parallel fast Fourier transform. Example simulations are compared with analytical, integral‐equation and spectral Lanczos decomposition solutions and demonstrate the accuracy of the scheme.

The effect of local distortions on time‐domain electromagnetic measurements

Based on the time‐domain integral equation, we derive expressions for the effect of an anomalous body close to the receiver or close to the transmitter on transient electromagnetic measurements. Similar to magnetotellurics, the distortion of electric fields at late times can be described by a constant distortion tensor relating the secondary electric field to the primary field components that would be obtained in the absence of the body. The distortion of a single electric field transient is a static shift only for particular configurations over a layered half‐space. In the general case, the perturbation is time dependent because the direction of the total electric field vector varies with time. The theory nicely explains spatial variations in electric field transients measured during a high‐redundancy long‐offset transient electromagnetics (LOTEM) survey over an underground gas storage site. An inversion example with synthetic data illustrates how distortion can be corrected. The elements of the distortion tensor are determined simultaneously with the model parameters. Ambiguity is reduced by a regularization of the distortion parameters. In the example, the background model is recovered well, even for the difficult case where only one transmitter is used. The distortion of the magnetic field time derivatives caused by bodies close to the receiver is proportional to the time derivative of the primary electric step response. The distortion is generally not limited to early times and cannot be neglected in general. Transmitter overprint effects resulting in static shifts of vertical magnetic field time derivatives may also be understood from the theory.

Coupling of the solar wind to temporal fluctuations in ground magnetic fields

A study of the coupling of the solar wind to temporal fluctuations in high–latitude ground magnetic fields is presented. The fluctuation measure considered is the absolute value of the north‐south horizontal field time derivative (∣dBx/dt∣) averaged over a 30–minute interval. The fluctuation level is predicted using a nonlinear mapping of solar wind plasma and field measurements. The predictability of ground magnetic field fluctuations is shown to depend on both local time and latitude. For lower polar cap magnetometer locations, the coupling is highest at the local times that have the highest average fluctuation level and lowest at the local times that have the lowest average fluctuation level. The highest level of coupling is at auroral‐zone latitudes in the post‐midnight sector. The relative influence of different solar wind inputs is shown to be highly dependent on spatial location.

A first attempt at monitoring underground gas storage by means of time‐lapse multichannel transient electromagnetics

Two successive transient electromagnetic surveys were carried out over an underground gas storage site in France. The idea was to monitor changes in the gas bubble from the differences in the data. If successful, the new methodology could help to reduce the number of monitoring wells and finally reduce costs. Preliminary 3D modelling indicated that the resistivity changes caused by movements of the gas/water contact should be detectable in the electric field transients provided that the signal‐to‐noise ratio is at least 100:1. The surveys were performed with the TEAMEX multichannel acquisition system, adapted from a seismics system. The highly redundant data were analysed by calculating the relative differences in the electric field transients. The differences were common‐midpoint‐sorted and spatially stacked. Another approach was the calculation of electric field time derivatives in a log–log domain, to eliminate static shift effects which are present in the data. Even though the data quality is excellent from a classical point of view, neither of the two approaches reveals changes in the data which might be caused by changes in the gas reservoir. In future applications to monitoring, transmitters and receivers should be installed permanently, and the transmitter input waveform should be monitored continuously, to avoid some of the problems encountered here. Moreover, the signal‐to‐noise ratio will have to be further increased by at least one order of magnitude.