Determination of the Radius of the Ring Current in the Earth’s Core According to the Data of the INTERMAGNET Network Observatories

Abstract

The geomagnetic dynamo is currently considered the most likely source of the Earth’s main dipole field. However, the radius of the current ring located in the Earth’s core is not reliably known. There are methods for indirectly estimating the radius of this current. Another method is proposed that allows one to indirectly estimate the radius of the current ring inside the Earth’s core based on measurements of the Earth’s magnetic field by observatories included in the INTERMAGNET network. The results of measurements taken on a day with low magnetic activity were compared using the least squares method with fields that could be created by ring currents of different diameters at the locations of magnetic observatories. The assumption was made that the ring current in the model used is located in the plane of the Earth’s equator with the center coinciding with the axis of rotation of the Earth. Estimates of the current radius in the range of 957–1595 km were obtained, which corresponds to the boundary between the solid and liquid cores of the Earth. These results can refine the model of the structure of the Earth’s core and Earth’s magnetism.

1. Introduction

Having created the “Magnetische Verein” in 1834 [1], Gauss and Weber initiated systematic observations of the Earth’s magnetic field. Despite the vast amount of accumulated data from many years of observations, the nature of the origin of the Earth’s magnetic field is still a mystery. At the moment, there is a clear agreement only that there are several independent sources of the magnetic field. After the discovery of the Earth’s liquid core and the creation of the theory of geodynamo, the division of the sources of the Earth’s magnetic field into four main ones became the most widespread:

-

The liquid core of the Earth, generating the main dipole field;

-

The lithospheric field created by magnetized rocks;

-

The external field created by currents flowing in the ionosphere;

-

Electromagnetic induction field created by currents of the crust and outer mantle [2].

Currently, there is no consensus on the mechanisms and structure of the geomagnetic dynamo. There are several theoretical models [3] capable of describing the processes of magnetic field generation, each of which is based on its kinematic models of fluid motion in the Earth’s core. There are papers suggesting other hypotheses of the occurrence of a magnetic field in the Earth’s core, such as the hypothesis of the presence of geothermic currents [4]. There are also other theories of the origin of the magnetic field of planets and stars [5].

The advent of low-orbit magnetic observatories has allowed us to obtain a lot of new and valuable data on the structure of the Earth’s magnetic field and create fairly accurate models of it, but for these satellites, ionospheric currents flowing at altitudes of 110 km turned out to be “internal” currents, which makes it difficult to separate the sources of the magnetic field by their relative contribution to the total field [6]. The ability, according to ground-based magnetic observatories measurements, to separate the fields created in the Earth’s ionosphere and inside the Earth turned out to be very important. Nevertheless, space missions have clearly shown the importance of gradient measurements of the magnetic field. The Swarm mission, started in 2013, launched three satellites [7], two of which fly on the orbit with an initial altitude of about 450 km, measuring the field gradient in the east–west direction, and the third satellite is 100 km higher above them, measuring the vertical field gradient. The data obtained during the Swarm mission made it possible to “create” 300 virtual observatories evenly distributed above the Earth’s surface [8].

Of course, it is impossible to solve the inverse problem of magnetometry in a general way. However, if magnetic observatories are located in a certain way in space relative to the current source and there are a limited number of variants of the topology of this current, then using the least squares method, it is possible to determine a variant of the topology of the field source that could create the magnetic field closest to the observed one [9]. In the seventies of the last century, 36 magnetic observatories were installed in Scandinavia to study auroral currents with distances between stations of 100–150 km and with their orientation in the north–south and east–west directions [10] with a Cartesian coordinate system common to all stations. Another example would be their 10 AUTUMNX magnetometer array in North America [11].

In [12], an attempt was made to solve the inverse problem of magnetometry using the IGRF-2005 magnetic field model. In this work, the sources of the magnetic field were represented as virtual magnetized prisms located in the Earth’s core. Only the Z component of the magnetic field was analyzed. Four significant inhomogeneities were identified that create the Canadian, Siberian-Asian, Australian, and negative South Atlantic global magnetic anomalies observed on the Earth’s surface.

In [13], based on the average monthly three-component data from magnetic observatories, the contributions to the Earth’s total magnetic field from sources located in the core, lithosphere, ionosphere, and magnetosphere were separated. The authors obtained the radius of the sphere in the Earth’s core equal to 2658.2 km. This radius roughly corresponds to the region of the liquid core located in the middle between the boundary of the solid core of the Earth and the outer boundary of the liquid core with the lower mantle.

In [9], which considers the dipole model of the magnetic field, data are provided that dipole sources of the magnetic field are detected at a distance of approximately 0.2 Earth radii, which is 1276 km.

The purpose of this work is to determine the radius of the source of the dipole magnetic field located inside the Earth, according to the data of the ground-based magnetic observatories of the INTERMAGNET network. We have proposed a method for determining the radius of a current ring using vector algebra and the least squares method. Although there may be many such rings and they may be of different diameters, the dipole magnetic field will be a superposition of the fields of ring currents.

2. Materials and Methods

The induction of the magnetic field of a current element at any point in space is described by the Biot–Savart Equation (1), explained in Figure 1.

$$dB={\displaystyle \frac{\mu }{4\pi }}{\displaystyle \frac{\overrightarrow{I}dl\times \overrightarrow{a}}{a^3}}, a\ne 0$$

(1)

Figure 1 shows the Cartesian XYZ coordinate system with the center located in the center of the Earth. The direction of the X axis corresponds to the direction of the prime meridian. In the plane of the equator (XOY), a ring current with a radius of R flows, creating a magnetic moment P_m directed along the axis of rotation of the Earth in the direction of the north pole. The magnetic observatory M is located at a point with polar coordinates (r, λ, φ), where r is the radius of the Earth, λ is the geographical longitude of the observatory, and φ is its geographical latitude.

The distance from the ring current element dl, which creates the field dB at the observation point, is indicated in Figure 1 as a. By integrating Equation (1) along the circle R, the total value of the field strength B at the observation point is obtained. For the case of a ring current, for example, formulas for calculating field components in polar coordinates are given in [14]. In this paper, formulas from [15] were used to calculate the components of the field strength at the observation point in Cartesian coordinates, which is more convenient for further calculations. The formulas used are given in Equation (2). A number of variables in the formulas have been replaced by the symbols from Figure 1 for ease of understanding.

$$\begin{gathered} B_x={\displaystyle \frac{{\mu }_{0}IRr\sin \phi }{4\pi }}\cos \lambda {\int }_0^{2\pi }{\displaystyle \frac{\cos \theta }{{\left(R^2+r^2-2Rr\cos \phi ·\cos \theta \right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}d\theta \\ \begin{array}{c}{B}_y={\displaystyle \frac{{\mu }_{0}IRr\mathit{sin}\phi }{4\pi }}\mathit{sin}\lambda {\int }_0^{2\pi }{\displaystyle \frac{\mathit{cos}\theta }{{\left(R^2+r^2-2Rr\mathit{cos}\phi ·\mathit{cos}\theta \right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}d\theta \end{array} \\ B_z={\displaystyle \frac{{\mu }_{0}IR}{4\pi }}{\int }_0^{2\pi }{\displaystyle \frac{R-r\cos \phi ·\cos \theta }{{\left(R^2+r^2-2Rr\cos \phi ·\cos \theta \right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}d\theta \end{gathered}$$

(2)

It follows from these equations that if the field is measured outside the radius of the current ring, but in the plane of the ring itself (z = 0), then the magnetic field vector will always be parallel to the axis of the current ring (Bx = By = 0). If the field is measured outside the plane of the ring, the magnetic field vector will deviate radially. The dominant contribution to the level of the field itself, and to the slope of the vector, will be made by the current flowing in the part of the ring that is closer to the measuring point $dB\propto {\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$a^2$}\right.}$. Thus, knowing the coordinates of the measuring point and the values of the magnetic field vector, it is possible to create an equation of a plane in space that will necessarily intersect near the ring with the current. Formula (2) is convenient because it relates the polar coordinates of the field measurement points, which usually specify the geographical coordinates of objects on the Earth’s surface, to the Cartesian coordinates of the magnetic field components, which make it mathematically simple to obtain the equation of the plane for a known point in space and the field vector. The equation of the plane for a point $O_1\left(x_1;y_1;z_1\right)$ and a vector $\overrightarrow{B}\left(B_x;B_y;B_z\right)$ will have the form (3).

$$B_x\left(x-x_1\right)+B_y\left(y-y_1\right)+B_z\left(z-z_1\right)=0$$

(3)

This is a linear equation with 3 unknowns. By making measurements at another observation point located outside the ring plane, you can create an equation for another plane. The line of intersection of these planes will pass near the current ring. By choosing the third observation point, a third plane will be obtained, and the combined solution of three linear equations with three unknowns will give a coordinate in space located at some distance from the current ring. A graphical explanation of the method is illustrated in Figure 2.

Figure 2a shows a ring with a current I, the vector of magnetic induction B₁, measured at the observatory O₁, located on the “plane 1” in which this ring is located. Figure 2b shows the intersection of “plane 1” and “plane 2”, which was constructed from the measurement of magnetic field induction B₂, measured at the O₂ observatory, located on the “plane 2”. Figure 2c shows the intersection of “plane 1” and “plane 3”, which was constructed from the measurement of magnetic field induction B₃, measured at the O₃ observatory located on the “plane 3”. Figure 2d shows the intersection of three planes and their intersection point. To solve systems of three linear equations with three unknowns, the Kramer (Gabriel Cramer) method was used.

The distance of this intersection point of planes from the edge of the ring will depend on the coordinates of the magnetic field measurement points relative to the ring with current and the radius of the ring itself. To evaluate the effectiveness of the proposed method, we will simulate the measurements that will be carried out by virtual magnetic observatories located on the surface of the sphere with a radius R. As a source of magnetic induction, we will consider a ring with a current I, which has a radius of 0.25R and is in the plane of zero latitude (in polar coordinates). By changing the coordinates of the observation points (longitude and latitude), we will estimate the distance of the resulting intersection point of three planes from the edge of the ring with current. This will allow us to optimally form triplets of real magnetic observatories to calculate the size of the ring current inside the Earth.

We denote the spherical coordinates of the base “virtual” observatory φ₀, λ₀, located on the surface of a sphere with a radius R equal to the average radius of the Earth 6380 km. The coordinates of the two additional observatories are shifted in latitude and longitude by an angle of Δ. The coordinates of the second observatory will be (φ₀ + Δ, λ₀), and the coordinates of the third observatory (φ₀, λ₀ + Δ).

Given different coordinates of the base observatory in latitude and different distances between the observatory, we calculate the coordinates of the intersection point of the three planes passing through the points with the coordinates of the “virtual” observatories and perpendicular to the magnetic field vectors calculated for these coordinates. The calculation results are shown in

Table 1. Coordinates of the calculated intersection points of planes for R_ring = 0.25R_Earth = 1595 km.

φ0(°)		0	5	10	20	30	40	50	60
Δ(°)	r/Δz
0.1	r (km)	4354	4408	4554	5001	5459	5817	6063	6221
	Δz (km)	0	−10	−76	−519	−1405	−2567	−3778	−4856
1	r (km)	4355	4418	4572	5023	5478	5831	6073	6227
	Δz (km)	0	−13	−86	−549	−1453	−2622	−3830	−4900
2	r (km)	4358	4431	4592	5049	5500	5847	6084	6235
	Δz (km)	0	−16	−98	−584	−1507	−2684	−3889	−4948
5	r (km)	4376	4480	4663	5133	5574	5903	6125	6263
	Δz (km)	0	−29	−137	−694	−1671	−2869	−4065	−5094
10	r (km)	4444	4595	4811	5298	5718	6017	6211	6328
	Δz (km)	0	−57	−215	−891	−1956	−3185	−4361	−5338
15	r (km)	4559	4756	5000	5496	5890	6157	6321	6415
	Δz (km)	0	−96	−311	−1109	−2256	−3509	−4664	−5586
20	r (km)	4727	4967	5233	5729	6093	6323	6455	6523
	Δz (km)	0	−145	−424	−1349	−2574	−3847	−4975	−5841

,

Table 2. Coordinates of the calculated intersection points of planes for R_ring = 0.5R_Earth = 3190 km.

φ0(°)		0	5	10	20	30	40	50	60
Δ(°)	r/Δz
0.1	r (km)	4675	4746	4932	5432	5851	6113	6256	6328
	Δz (km)	0	−16	−116	−733	−1809	−3048	−4206	−5152
1	r (km)	4676	4759	4953	5454	5867	6123	6261	6331
	Δz (km)	0	−20	−131	−773	−1864	−3103	−4253	−5189
2	r (km)	4679	4776	4978	5481	5885	6134	6268	6334
	Δz (km)	0	−26	−149	−818	−1924	−3164	−4306	−5230
5	r (km)	4703	4838	5063	5566	5947	6174	6293	6350
	Δz (km)	0	−45	−207	−958	−2107	−3347	−4465	−5352
10	r (km)	4790	4983	5237	5731	6070	6258	6351	6392
	Δz (km)	0	−89	−320	−1204	−2415	−3652	−4726	−5553
15	r (km)	4937	5180	5454	5925	6217	6366	6430	6452
	Δz (km)	0	−146	−453	−1465	−2730	−3957	−4988	−5755
20	r (km)	5151	5435	5717	6152	6391	6495	6527	6529
	Δz (km)	0	−218	−606	−1744	−3053	−4267	−5251	−5958

and

Table 3. Coordinates of the calculated intersection points of planes for R_ring = 0.9R_Earth = 5742 km.

φ0(°)		0	5	10	20	30	40	50	60
Δ(°)	r/Δz
0.1	r (km)	5850	6000	6211	6373	6373	6357	6354	6361
	Δz (km)	0	−110	−576	−2021	−3395	−4458	−5230	−5767
1	r (km)	5852	6022	6227	6376	6372	6357	6354	6361
	Δz (km)	0	−137	−631	−2088	−3450	−4499	−5258	−5786
2	r (km)	5861	6048	6245	6380	6373	6358	6356	6363
	Δz (km)	0	−168	−693	−2161	−3509	−4543	−5290	−5808
5	r (km)	5916	6139	6308	6399	6382	6365	6362	6369
	Δz (km)	0	−270	−878	−2374	−3684	−4674	−5384	−5872
10	r (km)	6100	6323	6433	6454	6415	6391	6384	6387
	Δz (km)	0	−464	−1188	−2712	−3960	−4882	−5534	−5976
15	r (km)	6383	6544	6584	6533	6468	6432	6418	6414
	Δz (km)	0	−678	−1500	−3034	−4222	−5080	−5680	−6078
20	r (km)	6763	6802	6761	6633	6538	6486	6460	6450
	Δz (km)	0	−910	−1813	−3345	−4473	−5272	−5822	−6179

. For ease of analysis, the coordinates of the points calculated in the Cartesian coordinate system with the Z axis pointing from the center of the sphere towards the north pole were recalculated into values $r=\sqrt{x^2+y^2}$ corresponding to the distance of the obtained point from the OZ axis, and Δz = z corresponding to the deviation of the obtained point from the equatorial plane. All distances are given in kilometers.

When performing modeling for the base station latitude φ₀ = 0°, the value φ₀ = 0.00001° was used. This was chosen because any stations located in the plane of the current ring have magnetic field vectors strictly parallel to each other, which means that for them, the equations of planes will be identical and there will be no intersection of planes. A small shift in the latitude of station coordinates eliminates this problem.

The third case, with the ring current radius of 0.9R-Earth, does not correspond to the geomagnetic dynamo theory. Nevertheless, it was modeled to reveal the possibilities of the method under consideration. The analysis of the results obtained in the modeling showed that the smaller the distances between the observatories and the closer they are to the equatorial plane, the closer the coordinates of the intersection points of the planes are to the edge of the ring with current. As the radius of the current ring increases, the intersection points of the planes also move away from the axis of the current ring.

Thus, if we form groups of three magnetic observatories each, from among the observatories included in the INTERMAGNET network, and select a day with low geomagnetic activity, we can obtain a set of intersection points of planes normal to the vectors of the field recorded by the magnetic observatories. Then, setting different radii of the virtual ring with current, we can calculate similar points of intersection of planes for the modeled ring currents. Using the method of least squares, it is possible to find such a radius of the ring current, at which the minimum discrepancy between the coordinates of the points obtained during modeling and those obtained from the results of the real measurement of the magnetic field strength will be obtained.

Before using data from real magnetic observatories for calculations, they must be previously modified. All magnetic observatories measure projections of the Earth’s magnetic field in local Cartesian coordinate systems that are unique to each observatory. The X and Y axes of these coordinate systems are located in the plane tangent to the surface of the geoid at the location of the magnetic observatory and are oriented in the directions to the geographical “north” and “east”, respectively, and the Z axis is directed to the center of the Earth (Figure 3).

To bring the local coordinate systems to the universal Cartesian coordinates, it is necessary to perform two rotations of the coordinate systems. First, rotate around the Y axis of the local coordinate system using the rotation matrix (4) by an angle of φ + π/2 in order to orient the local Z axis in a direction parallel to the Z axis of the general terrestrial coordinate system (Figure 3a).

$$R_y\left(\phi +{\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)=\left[\begin{array}{ccc}\mathit{cos}\left(\phi +{\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)& 0& -\mathit{sin}\left(\phi +{\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)\\ 0& 1& 0\\ \mathit{sin}\left(\phi +{\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)& 0& \mathit{cos}\left(\phi +{\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)\end{array}\right]$$

(4)

The second rotation must be made around the Z axis (Figure 3b) by an angle of −λ in accordance with the rotation matrix (5), so that the X and Y axes of the local systems become parallel to the corresponding axes of the global coordinate system.

$$R_z\left(-\lambda \right)=\left[\begin{array}{ccc}\mathit{cos}\left(-\lambda \right)& \mathit{sin}\left(-\lambda \right)& 0\\ -\mathit{sin}\left(-\lambda \right)& \mathit{cos}\left(-\lambda \right)& 0\\ 0& 0& 1\end{array}\right]$$

(5)

3. Results

To perform the modeling, the measurement results of the magnetic observatories included in the INTERMAGNET network were taken for one of the days in which there were no disturbances in the Earth’s magnetic field. The date chosen was 23 October 2005. The ap index, which characterizes the planetary amplitude of magnetic field variations for mid-latitude magnetic observatories, was chosen as the selection criterion for a day with low geomagnetic activity. The ap index values are shown in

Table 4. Values of the ap index on 23 October 2005.

UTC	0–3	3–6	6–9	9–12	12–15	15–18	18–21	21–24
ap	4	2	2	2	3	3	3	0

.

The year 2005 was not chosen for any special reasons, it was chosen because the data of this year were processed in order to study the dynamics of magnetic disturbances and synchronization problems of geomagnetic observatories [16]. The averaged daily components of the magnetic field were used for the simulation.

On the selected date, data from 98 observatories out of 154 included in the INTERMAGNET network were available. A review of the magnetograms revealed data omissions at 28 observatories, which were removed from the total number of observatories. In total, data from 70 observatories shown in

Table 5. List of magnetic observatories whose data were used in modeling the magnetic field source.

	Observatory 1			Observatory 2			Observatory 3
	Code	Latitude	Longitude	Code	Latitude	Longitude	Code	Latitude	Longitude
1	aaa	43.25	76.92	nvs	54.85	83.23	lzh	36.09	103.84
2	abk	68.36	18.82	ups	59.90	17.35	bel	51.84	20.79
3	aia	−65.25	295.74	trw	−43.27	294.62	pst	−51.70	302.11
4	asc	−7.95	345.62	vss	−22.40	316.35	mbo	14.39	343.04
5	asp	−23.76	133.88	kdu	−12.69	132.47	lrm	−22.22	114.10
6	bdv	49.08	14.02	fur	48.17	11.28	mab	50.30	5.68
7	bel	51.84	20.79	abk	68.36	18.82	nur	60.51	24.66
8	bfe	55.63	11.67	mab	50.30	5.68	aqu	42.38	13.32
9	blc	64.32	263.99	fcc	58.76	265.91	pbq	55.28	282.25
10	bmt	40.30	116.20	gzh	23.97	112.45	kny	31.42	130.88
11	bou	40.14	254.77	dlr	29.49	259.08	tuc	32.17	249.27
12	box	58.07	38.23	qsb	33.87	35.64	sod	67.37	26.63
13	brw	71.32	203.38	cmo	64.87	212.14	shu	55.35	199.54
14	cbb	69.12	254.97	res	74.69	265.11	fcc	58.76	265.91
15	clf	48.02	2.26	ebr	40.96	0.33	aqu	42.38	13.32
16	cta	−20.09	146.26	cnb	−35.32	149.36	asp	−23.76	133.88
17	dlr	29.49	259.08	tuc	32.17	249.27	fcc	58.76	265.91
18	dou	50.10	4.60	ebr	40.96	0.33	fur	48.17	11.28
19	ebr	40.96	0.33	spt	39.55	−4.35	dou	50.10	4.60
20	esk	55.31	356.79	dou	50.10	4.60	ebr	40.96	0.33
21	fcc	58.76	265.91	pbq	55.28	282.25	frd	38.21	282.63
22	frn	37.09	240.28	tuc	32.17	249.27	new	48.27	242.88
23	fur	48.17	11.28	wng	53.73	9.05	bfe	55.63	11.67
24	gck	44.63	20.77	abk	68.36	18.82	nur	60.51	24.66
25	gdh	69.25	306.47	naq	61.17	314.57	sjg	18.11	293.85
26	gui	28.32	343.56	mbo	14.39	343.04	sfs	36.67	354.06
27	gzh	23.97	112.45	lrm	−22.22	114.10	phu	21.03	105.96
28	had	51.00	355.52	ler	60.14	358.82	ebr	40.96	0.33
29	hbk	−25.88	27.71	her	−34.43	19.23	aae	9.04	38.77
30	her	−34.43	19.23	tsu	−19.20	17.58	hbk	−25.88	27.71
31	hrn	77.00	15.55	ngk	52.07	12.68	nck	47.63	16.72
32	hua	−12.05	284.67	sjg	18.11	293.85	teo	19.75	260.82
33	iqa	63.75	291.48	thl	77.47	290.77	gdh	69.25	306.47
34	irt	52.27	104.45	lzh	36.09	103.84	nvs	54.85	83.23
35	kak	36.23	140.19	mmb	43.91	144.19	gua	13.59	144.87
36	kdu	−12.69	132.47	asp	−23.76	133.88	cta	−20.09	146.26
37	kny	31.42	130.88	kdu	−12.69	132.47	kak	36.23	140.19
38	kou	5.21	307.27	sjg	18.11	293.85	vss	−22.40	316.35
39	ler	60.14	358.82	had	51.00	355.52	clf	48.02	2.26
40	lrm	−22.22	114.10	kny	31.42	130.88	kdu	−12.69	132.47
41	lvv	49.90	23.75	abk	68.36	18.82	nur	60.51	24.66
42	lzh	36.09	103.84	irt	52.27	104.45	gzh	23.97	112.45
43	mab	50.30	5.68	clf	48.02	2.26	wng	53.73	9.05
44	mbo	14.39	343.04	asc	−7.95	345.62	kou	5.21	307.27
45	mcq	−54.50	158.95	sba	−77.85	166.76	eyr	−43.47	172.39
46	mmb	43.91	144.19	kak	36.23	140.19	kny	31.42	130.88
47	naq	61.17	314.57	stj	47.59	307.32	gdh	69.25	306.47
48	nck	47.63	16.72	ngk	52.07	12.68	ups	59.90	17.35
49	new	48.27	242.88	tuc	32.17	249.27	mea	54.62	246.65
50	ngk	52.07	12.68	aqu	42.38	13.32	fur	48.17	11.28
51	nvs	54.85	83.23	aaa	43.25	76.92	irt	52.27	104.45
52	paf	−49.35	70.26	maw	−67.60	62.88	czt	−46.43	51.86
53	pbq	55.28	282.25	fcc	58.76	265.91	ott	45.40	284.45
54	phu	21.03	105.96	irt	52.27	104.45	gzh	23.97	112.45
55	ppt	−17.57	210.43	hon	21.32	202.00	frn	37.09	240.28
56	res	74.69	265.11	cbb	69.12	254.97	blc	64.32	263.99
57	sba	−77.85	166.76	mcq	−54.50	158.95	drv	−66.67	140.01
58	sfs	36.67	354.06	val	51.93	349.75	had	51.00	355.52
59	shu	55.35	199.54	brw	71.32	203.38	sit	57.06	224.67
60	spt	39.55	−4.35	had	51.00	355.52	clf	48.02	2.26
61	stj	47.59	307.32	kou	5.21	307.27	sjg	18.11	293.85
62	tan	−18.92	47.55	hbk	−25.88	27.71	czt	−46.43	51.86
63	teo	19.75	260.82	tuc	32.17	249.27	bou	40.14	254.77
64	thl	77.47	290.77	iqa	63.75	291.48	stj	47.59	307.32
65	thy	46.90	17.89	aqu	42.38	13.32	gck	44.63	20.77
66	tuc	32.17	249.27	mea	54.62	246.65	bou	40.14	254.77
67	ups	59.90	17.35	bfe	55.63	11.67	aqu	42.38	13.32
68	val	51.93	349.75	sfs	36.67	354.06	gui	28.32	343.56
69	vic	48.52	236.58	new	48.27	242.88	frn	37.09	240.28
70	wng	53.73	9.05	mab	50.30	5.68	fur	48.17	11.28

were used in the modeling.

Two more observatories were matched to each observatory to find the intersection points of the planes. The obtained intersection points are shown in Figure 4a,c,e. The coordinates of the intersection points in the figure are given in kilometers.

For comparison, Figure 4b,d,e show the results of modeling the plane intersection points for the magnetic field calculated for a current ring with a radius of 0.5 Earth radius located in the equatorial plane centered at the Earth’s center. The magnetic field magnitude was calculated using Equation (2) for a sphere with a radius of 6380 km.

As can be seen from Figure 4, the constellations of points obtained from real magnetic field measurements and those obtained from the ring current field modeling look similar in appearance. Since in the case of modeling the field from the ring current, we know the geometry of the current ring exactly, and the appearance of the “constellations” of intersection points in Figure 4a,c,e is very close in topology to similar “constellations” in Figure 4b,d,f, we can expect that the magnetic field source also has a ring structure.

To solve the problem of determining the radius of the ring current that formed the magnetic field closest to that measured by real magnetic observatories, we calculate the functional (6) for each triple of observatories:

$$∆r_i=\sqrt{{\left({Xobs}_i-{Xring}_i\right)}^2+{\left({Yobs}_i-{Yring}_i\right)}^2+{\left({Zobs}_i-{Zring}_i\right)}^2}$$

(6)

where Xobs_i, Yobs_i, Zobs_i are the coordinates of points obtained from real measurements of magnetic observatories, and Xring_i, Yring_i, Zring_i are the coordinates of points obtained from modeling the magnetic field of the ring current for the i-th triple observatories.

We calculate the average ($\Delta \overline{r}$) and root-mean-square (${\sigma }_{\Delta r}$) values of this functional for all 70 groups of observatories at different values of the radius of the ring current. The results obtained in kilometers are shown in

Table 6. Average $\left(∆\overline{r}\right)$ and root-mean-square $\left({\sigma }_{\Delta r}\right)$ values.

Rring	0.05RE	0.1RE	0.15RE	0.17RE	0.2RE	0.22RE	0.25RE
$\overline{\Delta r}$	730	727	724	723	723	723	725
${\sigma }_{\Delta r}$	650	651	653	654	657	658	662
Rring	0.3RE	0.4RE	0.5RE	0.6RE	0.7RE	0.8RE	0.9RE
$\overline{\Delta r}$	734	789	917	1154	1616	2064	2295
${\sigma }_{\Delta r}$	672	703	764	903	1602	2012	912

.

4. Discussion

The obtained coincidence of the topologies of the “constellations” of the intersection points of the planes calculated for the real Earth’s magnetic field and for the simulated ring current was expected. The radii of the ring current inside the Earth turned out to be unexpected. We expected to obtain a result close to that obtained in [12] based on the average monthly three-component data from magnetic observatories. The radius of the sphere of 2658.2 km obtained by these authors could not be so accurate in practice. The processes in the Earth’s core are more global and precision in tenths of a kilometer is out of the question.

Aksenov’s work [17] states that the radius of a toroidal ring current is equal to 1437 km. This value differs significantly from the data from the previous work. During the simulation, the difference between the outer and inner diameters of the torus was set to 3 km.

Using the least squares method, we planned to estimate the real boundaries of the inner and outer radii of the current ring, since the ring thickness of 3 km clearly does not correspond to the scale of the processes taking place inside the Earth. The results obtained allowed us to estimate the width of the ring current in the range from 0.15 to 0.25 of Earth radius, which is 957–1595 km. But these values are located in the boundary layer of solid and liquid nuclei. Researchers of the solid core structure previously drew attention to the presence of the innermost core with a radius of 300–600 km [18,19].

Probably, the division of the core into a solid core and a liquid core is not very strict. Perhaps, there is some “semi-liquid” part of the core. Doubts about the existence of a strict boundary between the solid and liquid core of the Earth have existed for a long time [20]. It was shown in [21] that geophysicists underestimated the viscosity of the boundary layer between the solid core and the liquid core. Recent advances in the emerging paradigm of the correlation-wave field [22] give hope that it will be possible to interpret the obtained seismograms more accurately and clarify the physics of the processes occurring at the boundary of the solid and liquid cores of the Earth.

Another possible interpretation of the obtained results could be the presence of current rings of various diameters located on the surface of a certain sphere. In this case, the maximum radius of the ring obtained will correspond to the radius of such a sphere in the plane of the equator.

For verification, we performed calculations for several more days, including days with low magnetic activity. However, qualitatively, the results did not change. The Earth’s main magnetic field, measured on its surface, is in the range of 30,000 to 60,000 nanotesla, depending on the geographical location of the measurements, which is significantly higher than the variations associated with magnetic storms (several hundred nanotesla), as well as daily Sq variations. Averaging measurements per day reduces the effect of variations. The presence of errors related to magnetic storms or diurnal variations can lead to a change of the inclination of the planes by no more than 0.01 radians, which can lead to errors in determining the coordinates of the intersection point within 0.01 Earth radius.

We also did not take into account the actual height of the magnetic observatories above sea level. Such a simplification could not cause any significant errors, as it leads to a displacement of the planes passing through the geographical locations of the observatories and, accordingly, the points of intersection of these planes by an amount not exceeding units of kilometers.

A copyright certificate was obtained for the developed software [23]. The text of the program can be obtained from the authors of the article.

5. Conclusions

The obtained estimates of the radius of the ring current inside the Earth, which is considered the most likely source of the dipole component of the Earth’s magnetic field, may provide new initial data for clarifying models of Earth’s magnetism.

Modern capabilities of computers make it possible to test various hypotheses about the geometric topology and intensity of currents inside the Earth, in order to identify configurations in which the calculated field most closely coincides with the observed field. The ring model used in the work was only a simplification to facilitate the demonstration of the proposed method.

Studying the dynamics of changes of the coordinates of the points of intersection of planes can help identify magnetic observatories, the quality of which should either be questioned, or it could provide additional information about the geophysical processes occurring in the depths of the Earth.