This report presents a method for constructing a simplified numerical description of the electric current distributions in the ionosphere and gap region based on dipole-aligned loop elementary currents (DALECs). A theoretical basis for DALECs is presented, along with a prototypical algorithm for constructing an elementary numerical DALEC. The algorithm is verified and validated by combining DALECs with an efficient Biot-Savart solver in order to estimate magnetic disturbance on the Earth’s surface. We examine (1) simple scenarios with known solutions and (2) hemispherical magnetic disturbance fields obtained from a state-of-the-art global geospace circulation model.

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Michael Wiltberger was supported in part by the National Science Foundation Independent Research/Development Program. We thank Benjamin Murphy, Jeffrey Love, Kristen Lewis, and Brian Shiro from the U.S. Geological Survey for their helpful discussions and reviews.

Multiply | By | To obtain |

Length | ||
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kilometer (km) | 0.6214 | mile (mi) |

three dimensional

dipole-aligned loop elementary current

dipole-aligned loop elementary current system

dipolar elementary current system

field-aligned current

global geospace circulation model

Lyon-Fedder-Mobarry

magnetohydrodynamic

magnetosphere-ionosphere cross coupler

spherical cap harmonic analysis

spherical elementary current system

spherical harmonic analysis

The magnetic field measured on Earth’s surface varies in intensity and orientation on different time scales. The longest scales, perhaps tens of months to millions of years, are associated with slow changes in the geodynamo that diffuse outward from the Earth’s liquid iron outer core, through the mantle and crust, and finally into the geospace environment (

Shorter time scale variations in the geomagnetic field, perhaps fractions of a second to tens of years, are primarily the consequence of dynamical changes in ubiquitous electrical current distributions in the geospace environment (

More recently, the direct impact of geomagnetic perturbations on ground-based technological systems has come to be appreciated. For example, the modern oil and gas industry relies on accurate estimates of local magnetic declination for directional drilling. At high latitudes, large magnetic storms and substorms can force the needle of a compass several tens of degrees away from its nominal direction (

An important challenge when mitigating such technological hazards is the sparse and uneven geographic distribution of magnetic observatories that are not generally located near infrastructure of interest. Interpolation can be used to address this issue, but naive two-dimensional interpolation based on nonphysical basis functions has been shown to perform poorly (for example,

Physics-inspired basis functions can be fit to data and used to fill gaps between observed magnetic fields in a self-consistent manner. Spherical harmonic analysis (SHA;

A comparable approach is to invert for equivalent ionospheric toroidal currents using a divergence-free spherical elementary current system (SECS;

These violations to Fukushima’s assumptions (1976) are often ignored with the argument that resulting ground magnetic disturbance is small (for example,

We propose a nuanced deviation from these latter GGCM studies based on a (magnetic) dipole-aligned loop elementary current system (DALECS). The concept of a dipole-aligned loop is not new, being first proposed by

Image showing Boström’s Type 1 (left) and Type 2 (right) current sheet loops.

Figure 1. Image showing Boström’s Type 1 and Type 2 current sheet loops.

Observational uncertainty and theoretical disagreements remain among space physicists concerning the detailed physics that might drive these electric current systems. See

The remainder of this report details the physical and mathematical theory underlying DALECs, and offers a software algorithm to construct a self-consistent grid of elementary Type 1 and Type 2 current sheet loops that can be used to simulate ground magnetic disturbance when combined with the Biot-Savart law. For verification, we reproduce some of the early results of

This work is consistent with U.S. Geological Survey Geomagnetism Program priorities (

Discretizing each Boström loop within a DALEC involves tracing dipole magnetic field lines. The required coordinate transformations can defy intuition and become computationally burdensome if not handled carefully. Fortunately, a magnetic dipole lends itself to constructing a right-handed orthogonal coordinate system with relatively straightforward analytic expressions to transform to/from standard spherical polar coordinates. This allows a simple loop to be constructed in magnetic dipole coordinates first, then transformed into more realistic spherical coordinates.

is the radius in spherical polar coordinates;

is the co-latitude in spherical polar coordinates;

is the longitude in both spherical polar coordinates and dipole coordinates;

parameterizes displacement perpendicular to a dipole field for constant,

parameterizes displacement parallel to a field line such that

Using equation 1, a loop can be defined as a dipole axis-aligned rectangle. Converting the position of discrete segments of this rectangle into spherical coordinates requires inverting equation 1, which in turn involves the solution to a nontrivial equation. A computationally stable version of this inversion is:

is an intermediate value used to solve equation 1 for

is an intermediate value used to solve equation 1 for

is an intermediate value used to solve equation 1 for

is an intermediate value used to solve equation 1 for

In addition to these positional transforms, discrete vector quantities constructed in dipole coordinates need to be converted to spherical coordinates. A directional vector component along a particular axis is the product of a scalar and unit vector associated with that axis. So, unit-vector transformations from dipole coordinates to spherical are required:

is an intermediate value used to obtain unit vectors aligned with dipole axes; and

signifies a unit vector along the specified axis.

So, for example, if a segment of a Boström loop has an associated current vector only parallel to the magnetic field line (that is, along the

Converting the length of a path between two points is often necessary. For arbitrary paths, this can be complicated, but if the path can be decomposed into segments parallel to each of the orthogonal magnetic dipole axes, there exist straightforward transformations into pathlengths in spherical coordinates:

is the pathlength parallel to unit vector

For the sake of completeness, we acknowledge that a Type 2 Boström loop is never a perfect rectangle in magnetic dipole coordinates and that its ionospheric segment follows a path along the ionospheric spherical shell with length

Using the geometric relationships discussed in previous sections, one can construct a closed current loop basis composed of both a Type 1 and a Type 2 Boström loop. To start, assign a current sheet density vector with both meridional and zonal components at a single coordinate on a spherical shell. To convert the zonal component of this current sheet density into a current, assume it is meridionally uniform and multiply by its meridional extent, that is,

These currents will remain constant around their respective Type 1 and Type 2 loops, so it is now simple to construct a loop of current vector segments in magnetic dipole coordinates: just define nonoverlapping segments of the rectangle, assign the constant current to each segment (taking care to assign the correct sign), and calculate the length of the segment as the vector difference in neighboring magnetic dipole coordinates. If current densities of each segment are also required, it is necessary to calculate a cross section, but this is also straightforward in magnetic dipole coordinates. We now have a well-defined set of (1) magnetic dipole coordinates of discrete segments of the Type 1 and Type 2 Boström loops; (2) an associated directional vector for each segment, scaled by the constant current; and (3) a length and cross section for each segment, all in magnetic dipole coordinates. All that remains is to transform these back into spherical coordinates using equations 2–7.

The Biot-Savart relationship is an empirical relationship that describes the magnetic field generated by an electric current that is constant in time (that is, the magnetostatic approximation). It is named after Jean-Baptiste Biot and Felix Savart, who discovered the relationship in 1820. It is consistent with Ampere’s original circuital law but without Maxwell’s displacement current correction. It predates Maxwell’s more complete electromagnetism equations by about 40 years. The Biot-Savart relationship is

is the three-dimensional location of magnetic field estimate;

is the three-dimensional location of differential current density element;

is the magnetic field vector at

is the magnetic permeability of free space;

is the differential current density vector at

is the differential volume/area/length associated with

Essentially, equation 8 formalizes the right-hand rule of thumb so often invoked to crudely describe magnetic field lines (represented by curled fingers) generated by a line current segment (represented by the thumb). It is actually an indirect solution to Ampere’s law (

A number of implementation details are left to the software designer, but the pseudocode presented in appendix 1 captures the most pertinent aspects of a Python module used to verify and validate the DALEC concept in the Verification and Validation section. With the noted exception of array operations, the algorithms are presented as linear procedures for the sake of algorithmic clarity. Undoubtedly, there are design choices available that could improve performance significantly.

At first glance, algorithms 1 and 2 in appendix 1 may appear to be largely similar; indeed, the numerous redundancies can be exploited to further improve performance. However, there are also key differences that users must not ignore. Their inputs are almost identical, except for the current sheet density of the input ionospheric segment (_{ϕ}_{θ}_{ϕ}_{θ}

With a complete two-loop DALEC constructed in spherical coordinates, the Biot-Savart relationship can be used to estimate its magnetic influence at any point on Earth’s surface. Although the Biot-Savart integral can certainly be represented in spherical coordinates (for example,

Thus far, we have described basic algorithms for generating a single (two-loop) DALEC (uniquely characterized by its latitudinal and longitudinal ionospheric sheet current density) along with the geographic radius, width, and length of this ionospheric sheet segment. From these calculations, it is possible to estimate the magnetic perturbation that would be produced at any point outside the DALEC’s three-dimensional (3D) boundaries. If we were to generate a second DALEC with a sheet current that aligned with one of the sheet currents of the first DALEC, the difference in their coincident FACs would equal the discrete divergence of the ionospheric segments. Finally, if we were to generate a regular grid of such aligned DALECs, it could faithfully represent the ionospheric current sheet density distribution expected from a physically self-consistent, if simplified, height-integrated ionosphere.

There will likely be noticeable discretization artifacts if the distance between a DALEC element and a point at which a magnetic perturbation estimate is desired is comparable to, or less than, the spacing between the discrete DALEC elements themselves. This is a common problem when the perturbation estimate is desired on Earth’s surface because the ionosphere’s altitude is roughly 100 kilometers, and grid spacing for the ionospheric segments is typically larger than this. If this is unacceptable for the intended application, a simple numerical approach may suffice: near those locations where synthetic measurements are desired, users can simply subdivide the original ionospheric segment into a grid of identical current density segments that only differ in their respective boundaries. We will refer to this collection of smaller DALECs as a “macro-” DALEC. The FACs internal to this macro-DALEC will cancel each other, but the sheet currents at the edges, especially the ionospheric sheet current, will now be discretized finely enough that the numerical artifacts should effectively disappear.

This numerical trick could easily become computationally intractable, or at least so slow as to discourage its use. However, a subtle aspect of the Biot-Savart relationship can be exploited to immensely accelerate calculations, at least after some otherwise computationally expensive precalculations are done, so long as the target locations for magnetic perturbation estimates do not change with respect to the DALEC. In this scenario, returning to equation 8, everything inside the Biot-Savart integral, except for the current density ^{3}

Furthermore, if one recognizes that all currents inside a DALEC are proportional to (that is, scale linearly with) the input ionospheric current density, it should be clear that the magnetic field perturbation obtained from Biot-Savart must also scale linearly with the input ionospheric current density. Therefore, one may simply set the input ionospheric current densities _{θ}_{ϕ}_{θ}_{ϕ}_{θ}_{ϕ}

In short, not only can we synthesize a macro-DALEC to arbitrary numerical precision by building it from smaller elementary DALECs, but we can also construct a global grid of elementary DALECs to self-consistently represent horizontal ionospheric currents, FACs, and equatorial current sheets. This is still an equivalent current system and not a representation of the true magnetosphere-ionosphere system, but it is much closer to reality than the purely toroidal ionospheric current systems mentioned earlier and matches perfectly the geometric and physical assumptions implicit to many GGCMs.

Finally, it must be acknowledged that both Type 1 and 2 Boström loops are undefined if one of the FACs falls on a pole where

Generally, verifying that an algorithm does what it is designed to do, correctly, is a worthwhile exercise. Because ours is a numerical implementation of a nominally physical system, this can be accomplished by reproducing certain analytic solutions.

^{6} amperes. The ionospheric segment of this toroid spans 65.5–69.5 degrees latitude before diverting current along the magnetic field lines. The y-axis scale in ^{6} amperes distributed evenly across 4 degrees longitude. Finally,

Graphs showing

Figure 2. Graphs showing meridional transect of ground magnetic response to Type 2 Boström loop toroid constructed from dipole-aligned loop elementary currents, ground magnetic response to zonally bounded segment of Type 2 toroid measured at its eastern edge, and ground magnetic response to inverse of zonally bounded segment of Type 2 toroid measured at its western edge.

Turning to ^{6} amperes flowing westward in the ionosphere. No analytic solution for the magnetic disturbance caused by this analog of a substorm current wedge exists, but we can compare our results with those obtained by the original authors.

Graphs showing

Figure 3. Graphs showing contours of eastward component of magnetic disturbance caused by a Type 1 Boström current sheet loop constructed to emulate a substorm current wedge, the northward component of magnetic disturbance caused by this Type 1 Boström loop, and the downward component of magnetic disturbance caused by this Type 1 Boström loop.

The results are antisymmetric (eastward) or symmetric (northward and downward) about 10 degrees longitude, which is the center of this zonal loop, so we drop the redundant western half of the magnetic disturbance distribution. The lack of symmetry in the north-south orientation is due entirely to the dipole nature of the magnetic field lines. The overall shape and intensity of these contours match those of

A similar comparison can be performed for a Type 2 Boström loop. In fact, the loop used to generate the results presented in

Graphs showing

Figure 4. Graphs showing contours of eastward component of magnetic disturbance caused by a Type 2 Boström current sheet loop with a 4×4-degree ionosphere segment centered at +67.5 degrees latitude, northward component of magnetic disturbance caused by this Type 2 Boström loop, and downward component of magnetic disturbance caused by this Type 2 Boström loop.

Having verified that our algorithm produces expected results to an accuracy limited only by the resolution of our discretization, the question remains: is our algorithm useful or, more importantly, is it reasonable to use it instead of simpler and less computationally expensive algorithms? We chose to validate the DALECs algorithm by combining it with LFM-MIX for the purpose of estimating ground magnetic disturbance caused by its current distributions. Briefly, this complicated magnetosphere-ionosphere model works as follows: given ionospheric Pedersen and Hall conductances generated empirically from FACs produced at the LFM model’s inner boundary, MIX solves the current continuity equation, then uses the calculated ionospheric electric potential as part of a low-altitude boundary condition for the LFM model’s MHD solution. This mostly self-consistent framework couples two physical modeling regimes with very different spatial and time scales, relying on a relatively simple geometric (dipole-field) mapping of the MHD inner boundary to ionospheric altitudes.

The LFM model does not output electric currents directly because they are not required for its MHD solution. (For the purposes of this study, MHD magnetospheric currents are not necessary, but for completeness, we note that these can be readily calculated from the curl of the local MHD magnetic field.) MIX does provide FAC densities that are mapped to the ionosphere; however, these are reproduced numerically by the divergence between neighboring DALECs, so we only use the MIX FACs for verification. As already noted, MIX also provides electric potential and conductance (Pedersen and Hall) distributions on a spherical ionosphere surface. When combined with Ohm’s law, these electric conditions enable a straightforward calculation of the horizontal ionospheric sheet current densities, which in turn constitute the ionospheric segments of a system of DALECs.

These horizontal current sheets are presented as two-dimensional projections onto the northern hemisphere of a spherical ionosphere in

Maps showing ^{2}, microamperes per square meter)

Figure 5. Maps showing electric potential and Hall horizontal ionospheric currents from the Lyon-Fedder-Mobarry–magnetosphere-ionosphere cross coupler global geospace circulation model, field-aligned currents and total horizontal ionospheric currents, and electric potential and Pedersen horizontal ionospheric currents.

The geomagnetic disturbance caused by the same 3D DALEC distribution summarized by

Maps showing

Figure 6. Maps showing vertical ground magnetic disturbance and horizontal ground magnetic disturbance due to full dipole-aligned loop elementary currents (DALECs), vertical ground magnetic disturbance and horizontal magnetic disturbance due to DALECs ionospheric segments, and vertical ground magnetic disturbance and horizontal magnetic disturbance due to DALECs field-aligned current and equatorial current segments.

Maps showing

Figure 7. Maps showing vertical ground magnetic disturbance and horizontal magnetic disturbance due to Hall currents, discrepancy in vertical ground magnetic disturbance and horizontal magnetic disturbance (that is, all Hall or Pedersen and field-aligned currents), and vertical ground magnetic disturbance and horizontal magnetic disturbance due to Pedersen currents.

In short,

This is not to say that some other divergence-free equivalent current distribution could not reproduce an observed geomagnetic disturbance pattern—we already know this is possible from

This report presented a theoretical basis for a magnetic dipole-aligned loop elementary current (DALEC) system. A single DALEC (consisting of a Type 1 and Type 2 Boström current loop pair) is not especially novel, as numerical implementations of this three-dimensional electric current system were used to model substorms as early as the 1960s. The novel contribution here was our construction of a regular grid of DALECs with coincident dipolar FAC elements that can effectively reproduce the divergence of electric current in the ionosphere in a physically consistent and numerically structured manner.

We also presented a template algorithm and verified that a DALEC system does indeed reproduce many expected theoretical results by implementing this algorithm in the Python programming language. We demonstrated how such a system can be integrated with the LFM-MIX GGCM to simulate FACs and horizontal ionospheric currents, thus allowing ground magnetic perturbations to be accurately estimated using the Biot-Savart relationship. The latter may, at first, seem unnecessarily redundant to those not familiar with GGCMs. However, because magnetohydrodynamic models do not generally track current density as a state variable (current continuity is assumed) and require only ionospheric electric potentials to establish their inner boundary conditions, they have no explicitly calculated currents.

Our DALECS algorithm Python implementation has been included with the LFM-MIX software package and has been used in studies that compare and assimilate ground magnetic observations with simulated geospace current systems. However, we note that our algorithm is very flexible and is only loosely coupled with the LFM-MIX GGCM codebase. It can easily incorporate output from any model capable of producing horizontally integrated ionospheric current sheet distributions (most do this) and, combined with a Biot-Savart integrator, can synthesize magnetic disturbance on Earth’s surface.

Recall from the Introduction section that much recent geomagnetism research has focused on the estimation of the geoelectric fields that drive geomagnetically induced currents in electrically conductive technological infrastructure. Although a gold standard toward this end might be to physically model the full electromagnetic response within the Earth to changes in geospace source currents (for example,

A relatively simple solution that should improve simulated geomagnetic disturbance, to zeroth order at least, is to assume a super-conducting sphere below Earth’s surface, thus allowing the use of so-called image dipoles. This was proposed and implemented by

Finally, to expand on this idea of linear regression, an empirical system of DALECs may easily be obtained by inversion of observed ground magnetic perturbations. This can be done in much the same way that

Recently,