Estimating the Ionospheric Induction Electric Field Using Ground Magnetometers

The ionospheric convection electric field is often assumed to be a potential field. This assumption is not always valid, especially when the ionosphere changes on short time scales T≲5 $T\lesssim 5$ min. We present a technique for estimating the induction electric field using ground magnetometer measurements. The technique is demonstrated on real and simulated data for sudden increases in solar wind dynamic pressure of ∼ ${\sim} $ 1 and 10 nPa, respectively. For the real data, the ionospheric induction electric field is 0.15 ± $\pm $ 0.015 mV/m, and the corresponding compressional flow is 2.5 ± $\pm $ 0.3 m/s. For the simulated data, the induction electric field and compressional flow reach 3 mV/m and 50 m/s, respectively. The induction electric field can locally constitute tens of percent of the total electric field. Inclusion of the induction electric field increased the total Joule heating by 2.4%. Locally the Joule heating changed by tens of percent. This corresponds to energy dissipation that is not accounted for in existing models.


Introduction
In this paper, we investigate the ionospheric induction electric field (E ind ) using a new technique based on ground magnetometers.When studying ionospheric dynamics the ionospheric electric field (E) is often assumed to be a potential field (E pot ) .This assumption can be very useful as it may simplify modeling efforts significantly.Techniques such as AMIE/AMGeO (AMGeO Collaboration, 2019;Richmond & Kamide, 1988) and Lompe (Hovland et al., 2022;Laundal et al., 2022) model E pot by ignoring E ind that otherwise is implied by Faraday's induction law ∇ × E = ∂ ∂t B) (Faraday, 1832).Similarly, E ind is almost always ignored in the ionospheric solvers used to account for the magnetosphere-ionosphere (MI) coupling in magnetohydrodynamic (MHD) simulations (e.g., Lyon et al., 2004;Merkin & Lyon, 2010;Tanaka, 2000).We present a technique for estimating E ind based on measurements of ground magnetic perturbation.Essentially, allowing E ind to be measured from ground.
Transient events (e.g., sudden commencements (SCs) or substorm expansions) can result in large changes in the magnetic field (B) on a timescale of seconds or minutes.When ignoring Faraday's law the mutual interaction between the electrostatic and inductive processes is neglected which can be important during dynamic events.Yoshikawa and Itonaga (2000) provide a detailed explanation of the inductive ionosphere, from an E,J perspective (Vasyliunas, 2012).Here, we give a brief explanation of the mutual interaction from an B,v perspective.Field-aligned currents (FACs), which close through the ionosphere via divergent Pedersen currents

10.1029/2023GL105443
Key Points: • A method for estimating the induction electric field using ground magnetometer measurements is presented • Locally, the estimated induction electric field can constitute tens of percent of the total electric field • The spatial pattern of ionospheric Joule heating is shown to be highly affected by the induction electric field, even during weak induction

Supporting Information:
Supporting Information may be found in the online version of this article.
when under steady state and uniform conductance, are directly influenced by changes in magnetospheric conditions, such as the opening of magnetic field lines and anti-sunward convection.These changes are communicated through shear Alfvén waves, bending magnetic field lines and inducing a flow of electrons perpendicular to the bend, that is, a divergent Hall current.This creates a rotational electric field (E ind ) , associated with what we term compression flow (E ind × B) , essential for redistributing magnetic flux and achieving a new steady state.In other words, under steady state and uniform conductance, the Pedersen current closing FACs only exist due to a pre-existing divergent Hall current.This process illustrates the coupling between electrostatic and inductive process in the ionosphere, further modulated by ionospheric conductance, which dictates the balance between reflected and transmitted Alfvén waves, as discussed by Southwood and Kivelson (1991), Yoshikawa and Itonaga (2000), and exemplified in simulations by Dreher (1997).Vanhamäki et al. (2005) investigated the inductive effect on the ionospheric electric field using realistic timedependent three-dimensional models of the high latitude ionospheric current system.They found that ionospheric self-induction is locally important with E ind reaching a few mV/m.Vanhamäki et al. (2006) presented a new technique for calculating E ind in a non-uniform conducting ionosphere.The technique utilizes the Cartesian elementary current system technique and requires E pot and Hall/Pedersen conductances as input.Vanhamäki et al. (2007) applied this technique to derive E ind for a westward traveling surge, Ω-band, and intensifying electrojet.They found that E ind can reach magnitudes of several tens of percent of the total electric field.Takeda (2008) simulated E ind associated with FACs with periods of 60, 10, 4, and 1 min and found that E ind had a non-negligible impact when the period of the FACs was 4 min or less.
In this study, we present a technique for estimating the ionospheric induction electric field based on ground magnetometer measurements represented with a spherical harmonic (SH) expansion and present examples of the associated ionospheric plasma flow.The purpose of the presented technique is to go beyond the assumption of a potential electric field in empirical modeling (e.g., AMIE and Lompe).Co-estimation of the potential and induction electric fields is desirable to understand the temporal evolution of the system.From a practical point of view, it also avoids the mapping of the induction electric field into the potential electric field.Additionally, by including the time-dependency of the system, the result becomes more constrained as subsequent time-steps are linked via measurements, increasing the overall information.However, the incorporation of this technique into pre-existing empirical modeling frameworks is outside the scope of the current study and will be addressed in future studies.
Our technique uses spatiotemporal variations in the magnetic field to infer compressional flow is analogous with studies of core flow using time-dependent models of Earth's main magnetic field (e.g., Finlay et al., 2020Finlay et al., , 2023;;Sabaka et al., 2020).Spherical harmonic models of Earth's core magnetic field can provide information about changes in the motion of liquid metal in the outer core through estimates of secular variation.This information can be used as boundary conditions in models of Earth's dynamo (Schaeffer et al., 2016).To the knowledge of the authors, it is the first time ground magnetometer measurements have been used to inform about the inductive component of the ionospheric electric field.However, Vanhamäki et al. (2013) solved Faraday's law based on the radial magnetic field to derive the induced electric field at Earth's surface.
In Section 2 we present a technique for deriving the ionospheric E ind from ground magnetic field perturbations.A more thorough derivation is provided in the Supporting Information.In Section 3, the technique is demonstrated using synthetic data from a coupled geospace model presented by Shi et al. (2022) and real ground magnetometer measurements during SCs.Section 4 discusses the results.

Technique
In this section, we describe how an estimate of the ionospheric induction electric field (E ind ) can be derived from the temporal derivative of the radial magnetic field ∂ ∂t B r ) below the ionosphere.A more in-depth derivation is provided in the Supporting Information.
The ionospheric electric field (E) can be decomposed into three scalar fields using the alternative Helmholtz representation (Sabaka et al., 2010), Here r is the radial unit vector and ∇ S is the angular portion of the ∇ operator.
The curl of the ionospheric electric field (∇ × E) on a spherical shell can be described by ∂ ∂t B r on the shell according to Faraday's law.By inserting Equation 1 into Faraday's law ∂ ∂t B r can be expressed in terms of W, The scalar field W can be represented with a SH expansion, ) are unknown, but can be expressed in terms of the SH coefficients (Sabaka et al., 2010), (4) In practice a m,B n and b m,B n can be determined by solving a linear inverse problem with magnetic field measurements on ground as input.The resulting SH coefficients should be determined using the ionosphere as their reference height.However, if the coefficients are determined with Earth's surface as their reference height they can simply be upward continued to the ionosphere.This detail is important as it defines the altitude of the spherical shell on which E ind will be determined.Only the radial magnetic field component can be upward continued to the ionosphere because it is continuous across boundary layers, unlike the horizontal components.
The horizontal part of E ind is given by the last term of Equation 1, (5) In the ionosphere, where the field-aligned conductivity is high, the electric field maps along the magnetic field making E ⋅ B = 0.This allows for the determination of E ind,r .However, E pot is typically unknown.By assuming radial magnetic field lines E ind,r = E pot,r and the compression flow is given as where br = r in the northern hemisphere.
Through the merger of the technique presented here and empirical modeling techniques of the ionospheric potential electric field like AMIE and Lompe E pot and E ind might be co-estimated.This will be the focus of future studies.

Results
Estimating the induction electric field (E ind ) requires a SH model of ∂ ∂t B r .In this section, we apply our method to two different cases of SCs.One model is based on ground magnetic perturbations from an MHD simulation while the other is based on real ground magnetometer measurements.

Synthetic Data Example
The synthetic data is based on an MHD simulation of an interplanetary shock carried out and analyzed by Shi et al. (2022).During this event, the solar wind dynamic pressure increases by approximately 10 nPa.The REdeveloped Magnetosphere-Ionosphere Coupler/Solver (REMIX) (Merkin & Lyon, 2010) is used to determine the ionospheric current and assumes that ∇ × E = 0.The reader is referred to Shi et al. (2022) for further details regarding the simulation.The ground magnetic perturbation is determined by computing a Biot-Savart integral over the ionospheric currents, FACs, and magnetospheric currents on an equal area grid with a 0.5°latitudinal resolution down to 0°latitude.We represent the ground magnetic perturbation using SHs, where the SH coefficients (a m,B n , b m,B n ) are determined by solving an inverse problem similar to Madelaire et al. (2022a) with the SH expansion truncated at n = 100.The SH expansion is only done for external sources as the synthetic data does not include ground induction.
Figure 1 summarizes the technique for estimating E ind , using synthetic data of the preliminary impulse associated with a SC. Figure 1a shows ∂ ∂t B r on ground.Figure 1b shows a recreation of ∂ ∂t B r using a SH model based on ground magnetic perturbation.A comparison between Figures 1a and 1b shows that ∂ ∂t B r is reproduced well by the SH model.Figures 1c and 1d compare the estimated E ind and the ionospheric potential electric field (E pot ) from the MHD simulation.Comparison between E ind and E pot are done with respect to the first of the two subsequent timesteps used to determine ∂ ∂t B r .We find that E ind reaches up to 3 mV/m which locally can correspond to tens of percent of E (E = E pot + E ind ) in the high latitude ionosphere.Therefore, E ind can have a significant regional impact.Figure 1e shows Joule heating associated with E pot (i.e., Σ p E 2 pot ) which is a result of maintaining the steady state current system.Figure 1f shows the difference in Joule heating when including E ind , that is, The difference can locally be tens of percent, both positive and negative.However, the total Joule heating above 50°latitude only increases by approximately 2.4%.The pins in Figures 1e  and 1f illustrate the steady state convection and compression flow (Equation), respectively, where B is the magnitude of a dipole magnetic field.The dipole magnetic field is determined using the first SH degree of IGRF-12 (Thébault et al., 2015).The flow illustrates the expansion/compression of magnetic flux necessary to change the ionospheric current system from one steady state to another.

Real Data Example
The SH model based on real ground magnetometer measurements was provided by Madelaire et al. (2022a) and is the product of a superposed epoch analysis of SCs.Madelaire et al. (2022a) presented 12 models determined by dividing the list of SCs presented by Madelaire et al. (2022b) into 12 groups based on the Interplanetary Magnetic Field (IMF) clock angle and dipole tilt angle.In this example, we use the model created for SCs during northward IMF and positive dipole tilt (Summer in the northern hemisphere).The model is based on 175 events, the majority of which experience solar wind dynamic pressure increases around 1-2 nPa.The much smaller pressure increases in this model compared to that used in Section 3.1 results in significantly smaller ∂ ∂t B r and E ind .The SH model includes a separation between internal and external sources.Both sets of SH coefficients are upward continued to the ionosphere and combined before deriving E ind .Furthermore, to assess uncertainty, 50 realizations of the model were created by resampling the events used as input.
Figure 2 shows a time series of ∂ ∂t B r and compression flow associated with the SH model, based on real ground magnetometer measurements, starting 2 min prior to the initial increase in SYM-H (Iyemori et al., 2010).Epochs are synonymous with minutes.Here, ∂ ∂t B r is the median across all 50 model realizations and the compression flow is the bias vector (e.g., Haaland et al., 2007) scaled with the median magnitude.The preliminary impulse appears in Figures 2a and 2b.The main impulse appears in Figure 2c, equatorward and with the opposite polarity of the preliminary impulse.Over the following 3 min (i.e., Figures 2d-2f) the main impulse expands along the flanks toward the nightside while increasing in strength.The compression flow is around 2.5 m/s with a standard deviation of around 0.3 m/s.Additionally, a large-scale southward flow appears shortly after the appearance of the preliminary impulse.

Discussion
We presented a technique for estimating the ionospheric induction electric field (E ind ) using measurements of magnetic field perturbation below the ionosphere.The technique links a SH representation of the temporal derivative of the radial magnetic field ∂ ∂t B r ) to a scalar field W representing E ind .In an example with synthetic data, we found that E ind reaches values of 3 mV/m (Figure 1d) which locally can correspond to tens of percent of the combined ionospheric electric field (E = E pot + E ind ) in the high latitude ionosphere.From estimates of E ind a compression flow of approximately 50 m/s was calculated (Figure 1b), which represents the necessary expansion/ contraction of magnetic flux to reach a new steady state.The total Joule heating above 50°latitude increased by approximately 2.4% while local changes were tens of percent (see Figures 1e and 1f).Inclusion of E ind in the calculation of Joule heating adds two terms, that is, Σ P E 2 ind and 2Σ P (E pot ⋅ E ind ) .Assuming E ind = E pot / 10 results in E 2 ind being 1% of E 2 pot .Meanwhile, the cross-term can contribute up to 20% of the Joule heating depending on the alignment of E ind and E pot .However, the cross-term can be positive or negative.It is, therefore, unclear how much it contributes to the total heating when integrated over the entire ionosphere.The contribution from the cross-term is illustrated in Figure 1f and leads to a significant difference in ionospheric energy dissipation during dynamic events compared to the steady state case, even when E ind is an order of magnitude smaller than E pot .However, the estimated value of 2.4% is specific for the synthetic case being studied as both the magnitude and spatial extent of the temporally varying magnetic field depend on several exogenous parameters.Furthermore, the background level of Joule heating can also vary.
The MHD simulation carried out by Shi et al. (2022), used to create the synthetic data example in Section 3.1, applied the ionospheric solver REMIX (Merkin & Lyon 2010) which assumes steady state.Therefore, the ionospheric electric field is a potential electric field since ionospheric selfinductance is neglected (i.e., ∇ × E = ∂ ∂t B = 0).We calculate ∂ ∂t B as the difference between two steady states for demonstration purposes.The combined ionospheric electric field (i.e., E = E pot + E ind ) no longer satisfy the current continuity (∇ ⋅ J = 0) ensured in REMIX and is fundamentally inconsistent.Furthermore, the rotational current associated with E ind in Figure 1d contributes to the ground magnetic perturbation.This leads to a secondary and weaker induction effect which subsequently leads to a third and so on and so forth.The infinite chain of opposing and progressively induction effects is naturally accounted for when using real data.However, the synthetic data still provide insight into the usefulness of the presented technique.The magnitude of E ind is similar to previous studies (Vanhamäki et al., 2005(Vanhamäki et al., , 2007)).
The presented technique was also used on a SH model of SCs based on real ground magnetometer measurements (Madelaire et al., 2022a).The retrieved E ind and compression flow is around 0.15 ± 0.015 mV/m and 2.5 ± 0.3 m/s (Figure 2), respectively.Additionally, the compression flow is dominated by a large-scale southward flow.This is consistent with an intensification of the magnetic perturbation from magnetospheric sources due to compression of the magnetosphere.The same intensification gives rise to a step-like increase in SYM-H during SCs (Madelaire et al., 2022b;Russell et al., 1994).A largescale flow is likewise present in the example with synthetic data, that is, Figure 1f.In Figure 3 the contribution from magnetospheric currents to E ind (i.e., Figure 3b) and the associated Joule heating has been separated from that of ionospheric currents and FACs (i.e., Figure 3c) for the synthetic example.Magnetospheric currents (e.g., magnetopause and ring current) produce, to first order, a uniform magnetic field in ẑ.At the poles, this corresponds to a weakening of the magnetic field, an azimuthal induction electric field (westward on the dayside), and a large-scale southward flow in the northern hemisphere.The induction electric field in the southern hemisphere points in the same direction but b points outward giving rise to a large-scale northward compression flow.Essentially, there is a large-scale equatorward compression flow at high latitude in response to rapid increases in solar wind dynamic pressure.Oppositely, there is a large-scale poleward compression flow in response to rapid decreases in solar wind dynamic pressure.
It is unclear how to interpret local changes in Joule heating due to the inclusion of E ind .Hesse et al. (1997) showed that E maps between the ionosphere and magnetosphere for ideal MHD, that is, including inductive terms.If that  Madelaire et al. (2022b).Epoch is synonymous with minute.The purpose of this figure is to showcase the estimation of E ind using a SH model that is based on real ground magnetometer measurements.Furthermore, the data includes contributions from magnetospheric sources that give rise to a large-scale southward compression flow.

Geophysical Research Letters
10.1029/2023GL105443 holds in reality it would lead to an asymmetric spatiotemporal evolution, for example, during SCs.However, Hesse et al. (1997) also showed that the mapping is non-trivial in the presence of parallel electric fields.Regardless of how E ind maps between ionosphere and magnetosphere the spatiotemporal evolution of dynamic events, for example, transient current vortices associated with the preliminary and main impulse of a SC and rapid compression/expansion of the magnetosphere, lead to significant local changes in Joule heating.The duration of these local changes can result in ion upflow but are unlikely to cause neutral upwelling (Strangeway, 2012;Zou et al., 2017) observed lifting of the F region ionosphere, large and transient field-aligned ion upflow, and prompt but short-lived ion temperature increase in the transition between the preliminary and main impulse of a SC using PFISR measurements.
There are significant differences in the magnitude of E ind and the compression flow (Equation) between the two models.The SH model provided by Madelaire et al. (2022a) is a product of a superposed epoch analysis based on a list of solar wind dynamic pressure increases (Madelaire et al., 2022b).The majority of the events in the list are not interplanetary shocks, and experience smaller pressure increases compared to what is often seen in case studies and MHD simulations (e.g., Fujita et al., 2003;Madelaire et al., 2022b;Moretto et al., 2000;Slinker et al., 1999) showed that the events, on average, contain increases of a couple of nPa.The interplanetary shock simulated by Shi et al. (2022) increased by approximately 10 nPa.The vast difference in the size of the pressure increase along with the smoothing associated with superposing multiple events leads to a ∂ ∂t B r in the order of 10 nT/min (Figure 2) compared to the 10 nT/s (Figure 1) seen in the MHD simulation.This is likely the explanation for the smaller compression flow.

Geophysical Research Letters
10.1029/2023GL105443 combining various measurements (e.g., conductance, convection, and ground/space magnetic field measurements), similar to AMIE/AMGeO (AMGeO Collaboration, 2019;Richmond & Kamide, 1988).However, the use of SECS in Lompe makes it ideal for regional analysis.In the future, we hope to remove the necessity of assuming steady state when using Lompe by implementing a scheme to co-estimate E pot and E ind using a technique similar to the one shown here.Looking ahead, preliminary work suggests that data from EISCAT 3D (Kero et al., 2019) will open possibilities for empirical modeling frameworks of 3D ionospheric currents.Such advancements will allow us to move beyond the limitations of an infinitely thin ionosphere model.We might, therefore, revisit our technique in the future in attempts to expand it into 3D.

)
Here (θ, ϕ) are colatitude and longitude (n, m), are the SH degree and order (a m,W n , b m,W n ), are the SH coefficients, and P m n ( cos(θ)) is the Schmidt quasi normalized Legendre polynomial.The coefficients (a m,W n , b m,W n

Figure 1 .
Figure 1.A summary of how E ind is determined based on synthetic ground magnetometer measurements from an magnetohydrodynamic (MHD) simulation (Shi et al., 2022), along with the compression flow and Joule heating.Panels (a and b) show ∂ ∂t B r from the MHD and Spherical Harmonic (SH) model, respectively.Panel (c) shows the magnitude of E pot and its orientation as pins.Panel (d) shows the magnitude of E pot with the orientation of E ind overlain.Panel (e) shows the Joule heating and plasma convection associated with E pot as a contour and pins, respectively.Panel (f) shows the difference between Joule heating associated with E pot and E = E pot + E ind as well as the compression flow associated with E ind .The purpose of this figure is to validate the SH models' recreation of ∂ ∂t B r as well as demonstrate the technique for estimating E ind .

Figure 2 .
Figure 2. Illustration of ∂ ∂t B and E ind × B 0 drift based on the Spherical Harmonic (SH) model provided byMadelaire et al. (2022b).Epoch is synonymous with minute.The purpose of this figure is to showcase the estimation of E ind using a SH model that is based on real ground magnetometer measurements.Furthermore, the data includes contributions from magnetospheric sources that give rise to a large-scale southward compression flow.

Figure 3 .
Figure 3.A decomposition of the contribution to E ind and associated Joule heating.Panel (a) shows the modification to the Joule heating when including E ind as a contour similar to Figure 1f with E ind superposed as pins.Panel (b) shows the contribution from magnetospheric currents while panel (c) shows the contribution from ionospheric currents and Field-aligned currents.