On the magnetic field in Earth’s interior

This paper discusses the possible distribution of Earth’s magnetic field in its three main layers. As a first step, using the IGRF-13 model, it was found that the magnetic field energy exterior to the liquid outer core (i.e., inside the mantle and the surrounding Earth’s atmosphere) is about 6.7×1018 J (for epoch 2010) while the field strength exactly at the Core-Mantle-Boundary (CMB) is about 0.42 mT (RMS value). These findings can be further considered as reliable reference values among the large dispersion of published data for the total magnetic field in Earth’s interior. Therefore, utilizing the layer breakdown reported in a pioneering paper (Glatzmaier and Roberts [8]) according to which the inner core includes about 10% of the total energy content, it was made possible to validate a previously reported figure (year 1949: 9×1020 J) regarding the total magnetic field energy of the Earth. But due to substantial deviation of the latter from a later reported figure (year 1980: 7×1021 J), this paper continues the investigation for the purpose of clarifying the issue. In this context, two inverse-cubic dipole models and the bar magnet (tiny and cylindrical) model, as well as the IGRF-13 standard were applied and compared in the Earth’s interior. Overall, under certain conditions these findings are successfully compared with literature reports, and eventually (after many assumptions) reached an acceptable compromise regarding the spatial distribution of the magnetic field, while a stimulus for further reflection to other researchers was provided. At the end, a question is posed and somewhat answered whether the available magnetic field energy is abundant to feed the motion of futuristic lightweight electromagnetic vehicles revived in recent literature.


Introduction
The dominating view on Earth's magnetic field, as taught in high-schools and colleges, is that it can be represented by a long bar magnet having its magnetic poles close to the geographical (Northern and Southern) poles [1,2].However, under these conditions, simple calculations related to the dipole magnetic moment ( » Ḿ 8 10 22 A.m 2 ) and Earth's average radius ( = R 6371.2 E km) show negative interaction energy of rather small absolute value ( »-Ẃ 3 10 bm 12 17 J) where the superscript 'bm' stands for 'bar magnet', which is quite different from the magnetic field energy in the atmosphere and interplanetary space ( » Ẃ 8 10 infinite 17 J), let alone the Earth's interior.The latter fact was the primary motivation of this study.A secondary motivation was to examine whether a portion of the available Earth's magnetic field energy could be consumed by future electromagnetic vehicles, which are currently being researched by others.
It is well known that the geomagnetic field varies in terms of the latitude j in such a way that its strength at the poles (∼60,000 nT) is almost twice that at the equator (∼30,000 nT) while a sinusoidal distribution occurs inbetween.These facts have sustained the theory of a gigantic bar magnet with two poles (the geomagnetic North and South poles) on Earth's surface [1].On the other hand, permanent magnetization cannot occur at temperatures above the Curie point (354 and 770 degrees Celsius for Nickel and Iron, respectively).Therefore, since the core of the Earth has a temperature of usually claimed 5,200 °Celsius, and is not permanently magnetized [2], the bar magnet model is not accurate but even if it was we would have a difficulty to calculate the self-energies of the poles.
In contrast, the concept of a dipolar field is more reasonable (than merely a bar magnet model) and the magnetic moment M is an overall scalar measure of the field (note that between the years 1980 and 2015, the magnetic moment M has decreased from 7.906 A.m 2 to 7.723 A.m 2 [3]).More precisely, in 1936, Bartels [4, p.  228] attributed the magnetization to a strong dipole (magnet with two infinitely strong poles, infinitely close together, with finite moment, which is the product of pole-strength m in A.m and pole-distance d in meters, i.e., = M md).
From the above discussion a few reasonable questions arise as follows.Is the magnetic dipole long (gigantic) or short?Can it represent the magnetic field and the associated total energy (such as 9 10 20 J or 7 10 21 J in Earth's exterior and interior) with adequate accuracy?These questions will be our main concern in this paper.
Today, the above concepts (of magnets and dipoles) have been replaced by the accepted theory of geodynamo according to which electric currents within Earth's liquid outer core produce the geomagnetic field (for details, the reader may refer to [5] and papers therein).Obviously, the rotation of the Earth about its axis (spin) is anticipated to play a significant role in the orientation of the magnetic field (the Coriolis effect), as the convection and motion in the outer core are influenced by the planet's spin.This results in a roughly north-south orientation of the magnetic field, which is why we have a magnetic north pole and a magnetic south pole, which are not exactly aligned with the geographic north and south poles.The main difference of the geodynamo model with the abovementioned (gigantic or short) magnetic dipole is that the former is related to electric currents distributed within the liquid outer core of the Earth while the latter dipole is represented by a thin (tiny) bar magnet (magnetostatics consideration).As the numerical results of this study show, the differences could be somewhat reduced if instead of a tiny dipole we alternatively consider a cylindrical bar magnet for which energy estimations are possible and depend on the aspect ratio length-to-diameter [6].
It should become clear that all the above models have difficulties and are still open to further improvements but also to criticism.More precisely, the bar magnet model (associated to the Gilbertian point of view of magnetostatics) suffers from (i) infinitely huge field intensity near the poles and (ii) the need of determining its self-energy, as we shall see later in section 9. On the other hand, the (magneto-hydro-dynamics) MHD geodynamo model is much more realistic but requires the solution of three-dimensional Navier-Stokes nonlinear partial differential equations (PDEs) which highly depend on the imposed boundary and initial conditions [7][8][9][10][11].Nevertheless, in this paper we shall show that the conclusions extracted from MHD geodynamo model are very useful to estimate and validate previous claims about the total Earth's magnetic field energy (see section 2.2 and section 6).
From the practical point of view, the knowledge of the Earth's magnetic field is important because its evolution is vital for the protection of life, the integrity of electronic devices, satellites, and so on (see, [12][13][14]).Shortly, the magnetic energy in the atmosphere and the interplanetary space continuously dissipates and is replaced by energy transferred from Earth's interior.This is attributed to the chain reaction's energy of nuclear fission of heavy nuclear isotopes (U-238, U-235 and Th-232), which is a main, and in a sense 'unlimited', source of geothermal energy of Earth [15][16][17].
A way to validate and justify existing simple geomagnetic models, at a certain degree, is to implement the IGRF-13 model which is based on many collected measurements from several observatories and satellites (Alken et al [18]).This model is of great assistance and can be used to calculate the field and also the total magnetic energy outside and inside the Earth (at least within the mantle).
The above knowledge about the physical mechanism of feeding the dissipated geomagnetic field is very important not only for geophysicists and geologists but also for other, more technological, engineering reasons as well.Actually, there are a number of older patents and funded reports recently published in peer refereed journals that the gradient of the Earth's magnetic field can induce useful forces, which could be used for a kind of propulsion (fortunately small) [19][20][21][22].In this context, the reader may have doubts about engineering claims that the geomagnetic field could also feed small electromagnetic vehicles for human transportation (of course, if such technology could ever evolve from today's prototypes into actual machines).A couple of critical questions could be as follows: What would happen if such a technology was widely disseminated?Would the compasses operate well, and could the abovementioned nuclear fission replace the continuously consumed geomagnetic energy?
The paper is structured as follows.Section 2 presents a critical collection of data from literature which will be used later.Section 3 presents the bar magnet model and the associated self-energies and interaction energy.Section 4 presents the dipole model.Section 5 presents the IGRF-13 model as well as an inverse-cubic model and then derives concrete numbers for the energy trapped inside the Earth's mantle and atmosphere.Section 6 utilizes the energy breakdown in Earth's layers proposed in [8] and tries to relate it with previously reported total amounts of energy.Section 7 tests the applicability of the IFGR-13 model inside the Earth.Section 8 tests the applicability of the dipole model and compares it with the already presented IGRF-13 model in the previous section.Section 9 presents the bar magnet model, starting from a tiny magnet of various lengths and terminating with a cylindrical magnet of various aspect ratios.Section 10 compares the three aforementioned models.
Section 11 tries to bridge the discrepancies between reported averaged values (in the outer core and the interface between the inner and the outer core) and the reported energy proportionality 1:10 between the inner and outer core; it attempts a synthesis of the aforementioned findings and makes working hypotheses for the Earth's core.Section 12 discusses the power to maintain the dynamo.Section 13 is a feasibility study regarding the utilization of the geomagnetic energy.Finally, section 14 discusses all the findings of this paper in brief.

Earth's layers and energy reports
In this section we shortly present the structure of the Earth in layers, a couple of useful literature reports regarding the total magnetic field energy, and some interesting data concerning the intensity (or strength) of the magnetic field (i.e., the magnetic induction B in [Tesla] units).Note that in contrast to the Earth's surface at which [nT] is the standard, in Earth's interior most reports use [mT] units, while older papers use Gauss units (1 G = 0.1 mT).Most of this material will be used in later sections of this paper.

Earth's structure
As usual, for the sake of easiness the Earth is considered as a perfect sphere of average radius = R 6371.2 E km.If we start moving from the Earth's surface toward its centre, we meet three main layers in the form of spherical rings, i.e. (i) the mantle, (ii) the outer core, and (iii) the inner core (a complete sphere), of which the corresponding depths and radii are shown in figure 1.Given the Earth's radius R , E if the symbol d stands for the corresponding depth measured from Earth's surface, the radius measured from Earth's centre will be = r R d.

E
The surface at depth = d 2885.2 km (equivalently, radius = r 3486 km) is called Core-Mantle-Boundary (CMB) and separates the liquid outer core from the mantle.On the other hand, the Earth's Inner-Core-Boundary (ICB), at = = d r 5155.2 km, 1216 km, is the site where the liquid outer core solidifies and the solid inner core grows.

Energy reports
The electric currents related to the geodynamo take place inside the liquid outer core thus it could be supported that the physical phenomenon is dominated by proper nonlinear PDEs (Navier-Stokes) of which only the numerical solution is possible.This means that simple algebraic laws are possible only outside this domain, i.e., within Earth's mantle as well as the infinite atmosphere and the interplanetary space.Regarding the solid inner core, only assumptions and seismological computations may be performed.In general, it is claimed that 'Despite its small volume (less than 1% of the Earth's volume), the Earth's inner core contains about 10% of the total magnetic field energy.' (see, [23]).In more detail, Glatzmaier and Roberts [8] claim that 'the total energy of the magnetic field within the inner core is usually no more than 10% of that within the outer core, and the total magnetic energy exterior to the outer core is usually less than 1% of the magnetic energy within the outer core'.
Regarding the total magnetic field energy, in 1949, Bullard [24] proposed the amount of 9 10 20 J, while later, in 1980, Vehoogen [25] proposed the figure of 7 10 21 J.The corresponding estimated maximum values of the magnetic field, extracted from their manuscripts as reported, are shown in table 1.
A recent study by Provatidis [26] proposed an easy way to determine the energy in the infinite space outside the Earth (atmosphere and interplanetary space), which was found about 8 10 17 J.The latter value is from 1125 up to 8750 times smaller than the total amount of magnetic field energy mentioned in the above paragraph and table 1 as well.Therefore, the magnetic field inside the Earth is obviously the biggest portion of the total energy amount.Note that, in order to proceed with calculations, we must first estimate the magnetic field energy inside the Earth's mantle, a figure which is still missing.This is accomplished in section 5.2.3.

Field strength reports
Regarding the intensity of magnetic field B in Earth's core, usually expressed in milli-Tesla (mT), the published data are widely dispersed.In 2010, Buffet [7] proposed the core-averaged field of 2.5 mT, while after a relevant interview with him Sanders [27] puts it close to the CMB.In the same year, Gillet et al [28] give the radial magnetic field at CMB equal to 0.3 mT and RMS field strength of 4 mT inside the Earth's core.Earlier, in 1995, Glatzmaier and Roberts [29, p.205] claim that 'the maximum field intensity usually occurs near the ICB and is typically between 30 and 50 mT' while for the CMB they claim the value of about 1.0 mT [29, p. 207].In the same year, Braginsky and Roberts [30, p. 69] claim that 'K we may say with some confidence that a geodynamo in which B 100 rms G (10 mT) can be maintained in Earth's core'.Later, in 2022, Landeau et al [31] refer to the propagation of hydromagnetic waves, called Alfvén waves, in the outer core.Due to the fact that their propagation velocity is proportional to the magnetic field B, the latter is possible to be estimated.Therefore, the field strength deep in the core can be estimated to ∼4 mT (previously reported by Gillet et al [28]), which is about 10 times the field strength at the core-mantle boundary (CMB), and it corresponds to a magnetic energy of ∼10 21 J inside the core [31].
All the above figures will be used and commended in the next sections.

Bar magnet (magnetostatic) models
At any point P of an axial (meridian) plane which passes through the northern and southern poles of a long and tiny magnetic dipole, the induced magnetic flux intensities B are shown in figure 2. Thus, a bar magnet that consists of S-and N-poles, of length l (poles separated by distance = d l 2 ), pole strength m and magnetic moment = M m l 2 , ( ) results in a magnetic field of the two concentrated poles, as follows: where m p » ´-4 10 0 7 H/m is the vacuum magnetic permeability.As already said, the same components (B N and B S ) are considered on any meridian plane (i.e., these components do not depend on the longitude l of point P, shown in figure 3).(see, figure 3).Given the magnetic moment = M m l 2 , ( ) by analogy to the electric dipole (see, [32, p.684]), the Cartesian components of the total field at any distant point P x y , ( )are given in terms of the Cartesian coordinates (x y , ) by: Regarding the total magnetic field energy, according to Griffiths [33] and Smythe [34], it comes out from the volume integral of the magnetic field density


thus has a positive sign.Furthermore, the two components of the magnetic induction form the total field (henceforth the subscripts '1' and '2' replace N and S, respectively), of which the main interest is the square of the vector's magnitude, i.e., º B B .
Here, we make use of the well known vector identity: Substituting equation (4) into equation (3) and then integrating over the entire infinite volume, we have: where the superscript 'bm' stands for 'bar magnet'.
For point-like poles, the first two integrals in equation (5 ), represent the self energy of the point poles, and are infinite.By analogy to electrostatics, the third integrand is called the 'interaction energy' and is given by: The abovementioned 'interaction energy' corresponds to the 'potential energy' proposed by the pioneering researchers Coulomb and Poisson, while the total energy includes 'something more'.Clearly, when the general equation ( 6) is applied for a magnetic dipole, we have thus the interaction energy becomes negative while the total energy W total bm in equation ( 5) is a positive figure.This is turn means that we cannot ignore the two self-energy terms involved in equation (5), and therefore we have to implement a kind of 'classical renormalization' (see [35][36][37][38]).By analogy to the electric dipole, the simplest way is to consider at each pole that the magnetic strength (i.e., the supposed 'magnetic substance') is uniformly distributed along the surface of a sphere of radius R which is centered at the corresponding pole.
Actually, the volume integrals of B 1  and B 2  (self energies in equation (5)) are infinite terms which are successively written by virtue of equation (3) as follows: A reasonable geometric constraint is that the above virtual spheres which carry the 'magnetic substance', each of radius R, should not be intersected thus we have  R d 2, / which implies: .0008 10 total 21 J (i.e., the total energy 7 10 21 J proposed by Verhoogen [25] plus the amount of 8 10 17 J in the atmosphere [26]) and = Ḿ 7.7460 10

The dipole model
The dipole model is practically the same as the previously discussed 'bar magnet' model with the major difference that the pole distance d is adequately small.In more detail, while the bar magnet model leads to singularities at its poles while it gives a bounded field at its mid-point, in contrast the dipole model leads to a fictitious singularity only at its centre while it gives regular values at the distant geographic poles (N and S).In conclusion, since the dipole model does not accurately represent the field at Earth's centre it cannot calculate the total field magnetic energy unless a certain sphere near the Earth's centre is excluded.
The dipole model is one of the oldest and at the same time accepted by the scientific community.As also discussed in section 3, when the observer is far from the centre of the dipole (say > r d 10 ) it is possible to make some approximations for the resultant vector thus to derive a single formula for the magnetic field.In the context of geomagnetics, this is called the magnetic central dipole (CD) model.After elaboration on equation (2), the magnetic field is proportional to the magnetic moment, M, according to the formula: where q is the co-latitude, i.e., the angle formed by the dipole's axis Oy (i.e., ON ) and the line OP which connects the Earth's center O with an arbitrary point P, while r is the length OP (see, figure 2).For this equation to be valid the magnet must be of very small length d, i.e. both poles should be close to Earth's center (so as the criterion > r d 10 is satisfied).Applying equation (10) at a point on Earth's surface (so, = r R E ) at the CD equator where obviously q = 90 0 degrees, i.e. q = cos 0, equation (10) determines the so-called 'reduced moment' B 0 (a reference field): 11 Then, dividing equations (10) and (11) by parts, the moment M is eliminated thus leading to where q B r, ( )is the measure B | |  of the vector B,  a sum of its radial and angular components.Note that on the Earth's surface the field q B r, ( )varies between B 0 and B 2 0 (peak value).As shown by Koochak and Fraser-Smith [3], the abovementioned B 0 is calculated using the first coefficient of the spherical harmonic expansion, i.e. in terms of first three Gauss coefficients used in the IGRF models: ( ) ( ) ( ) while the position angles are found by q = -g B cos , Based on equation (12a), the root mean square field (RMS) values in the CD model on Earth's surface have been analytically calculated and their values are cited in Provatidis [26].In brief, as also happens with other areas of physics, the RMS value of the field intensity on Earth's surface equals to the peak value ( B 2 0 ) divided by 2 , thus eventually giving (for a complete proof see [26]): ( ) the magnetic energy trapped in the spherical ring between radii r 1 and r 2 is given by:

RMS CD RMS
For the particular case in which the radius r 1 refers to Earth's surface ( = r R E

1
) while r 2 tends to infinity, the field energy trapped in the atmosphere and the interplanetary space becomes: Considering the variation of the magnetic moment M within the years 2000 and 2020, the abovementioned energy W infinite dipole is shown in the third column of table 2, while its ratio with the forthcoming IGRF-13 model (5 th column of table 3) is shown in the fourth column of table 2.

The IGRF-13 model
The International Geomagnetic Reference Field (IGRF) model is a set of coefficients of the potential V of the Earth's main magnetic field, which is published every five years, over which time the coefficients change as the Earth's main magnetic field changes (Alken et al [18]).The data have been collected from many observatories and satellites and then spline-approximation and least-squares fitting have been performed.The answer to the question regarding the Earth's area in which this model is applicable is not very clear.While the original paper [18] avoids defining the applicability of the IGRF-13 model, in the corresponding official website (see, [39]) it is mentioned that this model 'is used widely in studies of the Earth's deep interior', but without explaining what exactly the deep interior is.However, it is a common secret than IGRF-13 can be safely used outside the coremantle-boundary (CMB), i.e., within the mantle as well as in the atmosphere and the interplanetary space.Of course, any reader can understand that this ambiguity is because no direct measurements can be collected from that depth, thus we resort only to indirect calculations.

State-of-the-art on magnetic field
Johann Carl Friedrich Gauß included the method of spherical harmonic potential to the geomagnetism from the potential theory in 1839.This method obtains the scalar potential for geomagnetic field from the solution of Laplace's equation.International Geomagnetic Reference Field (IGRF) is the negative spatial gradient of the scalar potential as well.It can be written as where V is the scalar potential for geomagnetic field, which is rewritten as Earth's radius.The r, q and l are the geocentric coordinates, r is the radius, q is the co-latitude (q = 90 o -latitude), and l is the longitude.The coefficients (g n m and h n m ) are the well known Gaussian coefficients (which have been determined through the least-squares method) put forth by the International Association of Geomagnetism and Aeronomy (IAGA) for the IGRF, and q P cos n m ( ( ) represents the Schmidt quasi-normalized associated Legendre functions of degree n and of order m.
Then for any point with geocentric coordinates r, q and l, the three components of the geomagnetic field (B , r q B and l B ) can be analytically determined as follows: The most recent values of Gaussian coefficients (i.e., those of version 13) may be found in Alken et al [18] while the previous ones in Finlay et al [40].Recently, a MATLAB ® computer code for calculating the numerical values of the abovementioned field components ( ) according to IRGF-13 was released by Compston [41].It is noted that the n-th harmonic includes )terms, while the maximum number in the series expansion is = = n k 13 max harmonics.Moreover, it is well known that the energy density u of the geomagnetic field is given in terms of the current magnetic field ( ) [ ] / through the relationship (see also, equation (3)): where, m p = ´-4 10 0 7 N/A 2 , is the magnetic permeability in a classical vacuum (here is supposed to be the air which surrounds the earth).
Substituting equations (16)-( 18) into equation (19) and further integration over the differential volume element u q q l = = S r r r ) of the atmosphere, considering the orthogonality properties of the Legendre functions, gives the analytical formula for the total energy u = W ud , ∭ which is presented in section 5.2.

Energy between Earth's surface and arbitrary altitude
For any altitude, h, the energy trapped between the Earth's surface ( = r R E

1
) and the sphere of radius ) can be set in the form ( [17], [42], instead of the approximate equation (13a)): Equation (21) shows that each harmonic has its own decay, starting from r 1 3 / ( = n 1) and ending at r 1 27 / ( = n 13).The absolute value in equation (21) has been set so as to cover the case of negative altitudes ( < h 0), i.e., the positions inside the Earth.Note that the symbol W 12 of equation (20) differs from W bm 12 in equation (6).

Energy in the infinite space around the Earth
According to [43,44], the total energy between the Earth's surface and the end of the infinite space is given by: Actually, equation ( 22) can be derived immediately from equation (21), simply setting  ¥ h .Obviously, equation ( 22) can be written as the contribution of n-harmonics to the whole energy (in Joule),

The root-mean square (RMS) value
The root-mean square (RMS) value of the main field q l B r, , ( )on Earth's surface S of radius R E is defined by the ratio: 24 The quantity B RMS E , represents a virtual state of uniform field over the Earth's surface with the same energy nearby as the real one (in our study is well approximated by the IGRF model).Repeating the mathematical procedure according to which equations (20)-( 23) have been derived considering the orthogonality of the involved Legendre functions, the RMS field can be expressed in terms of its k harmonics as follows: In terms of statistics, the RMS value can be alternatively determined using the formula: where N is a large number of sampling points uniformly distributed on Earth's surface.
Practically, the RMS value is a sort of homogenization of the variable field B on the spherical surface of radius r (concentric to Earth's surface).This uniform variable incorporates the variation due to the latitude f and longitude l, thus we have to deal with a single number on each spherical surface.Obviously, when ¹ r R , E the area pR 4 E 2 in the denominator of equation (24) has to be replaced by the term pr 4 . 2 It is worthy to mention that all the thirteen harmonics ( = k 13) of the IGRF model contribute to the formation of the magnetic field B (min, max, and RMS values) on Earth's surface as well as its associated energy W IGRF infinite ( ) beyond the Earth's surface (i.e., in the atmosphere and interplanetary space).The same holds for any other spherical surface concentric to the Earth, at least between the CMB and Earth's surface.Note that the involved coefficients slightly depend on the epoch we study.
Using all the above equations, table 3 shows the evolution of the field and energy for the last two decades.In more detail, the minimum and the maximum value of the field B on Earth's surface (tabulated in the second and third column of table 3) are based on 4098 uniform points on a spherical model at which calculations were performed using the software released by Compston [41].The previous computations at these 4098 points have also been used in equation (26), and in this way have validated the identical results of the fourth column (labeled as RMS value) obtained by the straightforward formula of equation (25).An additional validation of equation ( 25) has also been performed in terms of equation (24) in which Gaussian quadrature has been conducted.In table 3 one may also observe that the two extreme values as well as the RMS value do not decrease at the same degree.For example, the average decrease of the minimum field value is 163 nT per 5 years, of the minimum value is only 37 nT, whereas the RMS value monotonically decreases by about 75 nT every five years.It is also worthy to note that the minimum IGRF-based value is smaller while the maximum one is greater than those (30,000 nT and 60,000 nT) reported by Kono [16].It is also worthy to mention that the ratio of maximum to minimum field values is about which is greater than the ratio in the CD-model which was previously discussed in section 4.
Equation (22) has been used to determine the magnetic field energy in the atmosphere and the interplanetary space (fifth column) while equation (20) for the field energy inside the Earth's mantle (sixth column).Of major importance is the seventh column of table 3 which summarizes columns 5 and 6 thus illustrating the total energy beyond the CMB (i.e., outside the core).In brief, the magnetic energy outside the core according to IGRF leads to a total of (∼6.68 × 10 18 J).The latter figure includes all the 13 harmonics of the IGRF model and is within the limits ´´  5 10 6.68 10 5 10 18 18 19 J, which have been predicted by Verhoogen (1980, pp. 71-73).The importance is because pioneering papers such as [8] express the energy in the inner and outer core as a percentage of the aforementioned value (∼6.68 × 10 18 J), which however had not been definitely reported in a reliable way so far.

Alternative inverse-cubic law 5.3.1. General
As has been previously discussed by Provatidis [26], the abovementioned quantity B RMS E , can be easily used as a reference value to determine the RMS value over any radius r, exterior or interior to the Earth, using the following inverse-cubic law:

Outside Earth's surface (atmosphere and interplanetary space)
The adoption of the formula for Earth's interior in conjunction with equation ( 27) leads to the following closed-form expression: where R in is the radius in the interior of Earth ). Mathematically, equation (28a) is not applicable at the center of the Earth where ( ) / everywhere, particularly when  r 0, would lead to the unrealistic value  ¥ B RMS at this.
For the particular case in which the depth d becomes negative and of a very large absolute value, the denominator in the bracket tends to infinity, thus the field energy trapped in the atmosphere and the interplanetary space is approximated by:

Earth's mantle
If we select the depth at = ´= d 0.5 2885.2 km 1442.6 km, i.e., at the middle of the mantle, the evolution of the magnetic energy between the years 2000 and 2020, based on the IGRF-13 (second column) model and the inverse-cubic law (third column), is shown in table 4. One may observe the slight differences between these two models (equation ( 27) underestimates the IGRF-13 model by about −4%).
If we further choose = d 2885.2 km, i.e. exactly the depth of the CMB, the same results are presented in table 5. Now, one may observe that the difference between the two models is larger than previously, i.e., equation (27) underestimates the IGRF-13 model by −28.4% (ratio ∼1.40).

Earth's core mantle boundary (CMB)
Regarding the CMB which by definition is the utmost inner boundary of the mantle, the corresponding full IGRF-based results (13 harmonics) for the year 2020 are shown on the left of table 6.One may observe that the minimum value decreases in a non-uniform manner while the maximum ones in a more monotonic way.Dividing the third by the second column in table 6 we derive the fourth column from which one may conclude that, in contrast to the Earth's surface, the calculated values on the CMB lead to large ratios (maximum to minimum field value) which vary between 48 (years 2005 and 2010) and / (for the year 2020).
Note that the latter was estimated using a cloud of 4098 points uniformly distributed on the surface of the sphere at the CMB.
Regarding the RMS value of the field B at CMB, one may observe that it monotonically decreases with time, in both models (IGRF and Inverse-cubic).One may also observe that the inverse-cubic model (equation ( 27)) underestimates the RMS value by −35% with respect to the IGRF standard.In addition, table 6 shows that the RMS value of the field B in the IGRF-13 model (equation ( 25)) is a little larger than 0.41 mT.However, to match with energy content mentioned in older publications (years 1949 [24] and 1980 [25]) henceforth we have rounded it at 0.42 mT.

Interpretation of the energy amount in the two inverse-cubic models
Although in equations ( 27)-(28a), (28b) we have considered the RMS value of the field on Earth's surface which reflects the same energy on it with that in the IGRF-13 model, the global adoption of the inverse-cubic ~r 1 3 ( ) / -law leads to a larger infinite energy.To make this point clear, let us focus on a sphere of radius r with > r R , E on which the accurate (IGRF-based) RMS-value (similar to equation (25)) is proved to be given by: Equation (29a) denotes that for each modal number n the radial decay of the magnetic field is different, as it goes with / Furthermore, equation (29a) may be equivalently written as follows: It is easy to validate that when = R r 1 E / the square root in the right-hand side of equation (29b) becomes equal to the quantity B RMS E , (cf equation ( 25)).Moreover, outside the Earth (where < R r 1 E / ), elementary algebra dictates that this square root becomes smaller than the term B RMS E , and thus we have proved the following inequality: Therefore, equation (30) clearly shows that at any radius r outside the Earth (i.e., for

Modified RMS value and associated modified dipole moment
While equation (25) expresses the situation on the Earth's surface only, if we demand that B RMS r , decays according to (27), i.e., inversely proportionally to the cube of the observer from the Earth's center (~r 1 3 / ), by analogy to (28b) the approximate total energy in the infinite space outside the Earth will be: If we demand that the above approximate energy is exactly equal to that of the full IGRF model, after manipulation the assumed equal left sides in equations (22) and (31) lead to the following second candidate RMS-value on the Earth's surface: Clearly, when (32) is substituted into (28b) we get the same infinite energy with that of the IGRF model (by definition).
It is also worthy to mention that the relationship between the abovementioned ¥ B E ( ) in (32) and a modified magnetic moment ¥ M will be similar to that of equation (11) with Therefore, the adoption of (32) in conjunction with (33), for the year 2020 gives and an associated modified dipole moment = ¥ M 7.8585 10 22 A.m 2 , which lead again to exactly the same energy = Ẃ 7.9595 10 J. ).In this sense only, if the contribution from a centred dipole is subtracted from the observed field, the residual (non-dipole field) contributes by 7%, otherwise (with reference to = Ẃ 7.9595 10 J infinite 17 ) the contributions of dipole and non-dipole terms become 96.2% and 3.8%, respectively.

Solving the inverse problem
Having determined the energy outside the CMB in section 5, now we can proceed with estimations on energy breakdown in Earth's interior.Based on the general conclusions of Glatzmaier and Roberts [8], in this section we shall derive the total amount of energy, which according to Bullard [24] is estimated to 9 10 20 J. = Ẃ Energy of magnetic in atmosphere: 8 10 J.
infinite 17 Adding the abovementioned values of 8 10 17 J for the atmosphere, and 5.9 10 18 J for the mantle (see, table 3), we eventually obtain the amount of ~6.7 10 18 J, which is used as a basis of reference.Actually, as also mentioned in section 2.2, in their excellent paper Glatzmaier and Roberts [8] have shown that 'the total energy of the magnetic field within the inner core is usually no more than 10% of that within the outer core, and the total magnetic energy exterior to the outer core is usually less than 1% of the magnetic energy within the outer core'.
We hypothesize that saying 'usually' in the above sentence, Glatzmaier and Roberts [8] mean that is what they estimate in multiple simulations.We would say these values are highly uncertain and depend a lot on initial boundary conditions.It is not a secret that based on different initial boundary conditions different results can be obtained for the dynamics within the outer core, including the inner core differential rotation, etc.
In any case, if W , inner core W , outer core W mantle and W infinite are the energies trapped into the inner core, the outer core, the mantle and the infinite space, respectively, we have: where W total is the unknown total energy of Earh's magnetic field, with the following restrictions: Clearly, the inequalities in equations (35) and (36) come from the above quoted text in italics, as claimed by Glatzmaier and Roberts [8].
Moreover, if we consider the known volumes (V , inner core V , outer core and V mantle ) of the three layers in Earth's interior, and also assume for each of them that the corresponding energy is proportional to the square of the field strength (i.e., their RMS values B , inner core B , outer core and B mantle ) according to equation (19), (34) becomes: Therefore, the above calculation of the total Earth's magnetic field energy is shown in table 7(a), which is » Ẃ 7.44 10 total 20 J, i.e., somewhat less than the figure of 9 10 20 J which was reported by Bullard [24].Nevertheless, since the energy exterior to the outer core is allowed to be less than = p 1%, if we preserve the relationship 1:10 for the inner and outer cores, respectively, and further reduce the percentage from 1% to ¢ = p 0.825%, then table 7(b) shows that we obtain exactly the figure of 9 10 20 J proposed by Bullard [24].A most accurate calculation should consider the decrease of the figure of 9 10 20 J, from the Epoch 1949 (to which Bullard's [24] calculations refer) to 2010, where the magnetic moment was reduced from = Ḿ 8.068 10 1950 22 A.m 2 (see, [45]) to = Ḿ 7.7460 10 2010 22 A.m 2 .If we do so, considering that the energy is proportional to the square of the magnetic moment M, the amount of 9 10 20 J should reduce to about 8.25 10 20 J.This means that in table 7 the percentage of the energy (p) outside the CMB (mantle plus atmosphere) is somewhat between ¢= p 0.825% and = p 1%.For the sake of completeness, in table 7(c) we repeat the same calculations for the percentage of ¢¢ = p 0.1054% and then the figure of = Ẃ 7 10 total 21 J, proposed by Verhoogen [25], is derived.In other words, the determination of the percentage in the claim by Glatzmaier and Roberts [8] that 'Kis usually less than 1%...' is a key-point issue to accurate estimate the total Earth's magnetic field energy.
Another base of reference to calculate the total energy is the averaged field energy density.Therefore, considering that the Earth's volume is  1) is only some extractions from the papers [24,25].In table 8 one may observe that Bullard [24] and Verhoogen [25] have probably rounded the whole Earth's averaged field at about 15 G (1.5 mT) and 40 G (4 mT), respectively.
Furthermore, based on the values shown in table 7(b), the breakdown of energy density and the corresponding averaged field according to equation (19) -for the total energy estimation by Bullard [24]-are shown in table 9.One may observe that the fourth column of table 9 suggests that the field strength is 3.5 mT for the outer core, which is close to the above reported value of ∼4 mT (Gillet et al [28]).Nevertheless, the value of 5.2 mT in the inner core is very small compared to the value of 10 mT reported by Braginsky and Roberts [30] as well as the values in the interval [ 30 50] mT reported by Gillet et al [28] as well as Glatzmaier and Roberts [29].
Finally, the energy density and the averaged field -for the estimations by Verhoogen [25]-are shown in table 10.In the fourth column, one may observe that the RMS value of 9.70 mT within the outer core is 2.4 times the value of 4 mT reported in [28], and 3.9 times the value of 2.5 mT reported in [7].Regarding the inner core, the findings of 14.56 mT is 46% higher than the claimed value of 10 mT (100 G) reported by Braginsky and Roberts [30], and is outside the interval  mT reported by [28].
In other words, based on a simple rule of percentages in Earth's layers reported by Glatzmaier and Roberts [8], and having determined a concrete content of energy in Earth's mantle and atmosphere ( 6.7 10 18 J), it was possible to determine possible field strengths and corresponding energy breakdown.Having said this we also have to point out that not all the field conditions at the ICB and the averaged value in the outer core have been fulfilled.Overall, based on the published data so far, we have not adequate confidence on the total amount of magnetic field energy, which highly influences the value of the field.
To conclude, it is suggested that previous models and particularly the three-dimensional numerical solutions of the MHD dynamo model, to be readjusted with respect to the reference energy amount of 6.7 10 18 J outside the CMB and RMS field 0.42 mT on CMB; both figures can be safely taken for granted.In addition, a careful post-processing of the results in the MHD dynamo model could reveal the accurate distribution of the field intensity (RMS value B r ( )) in the radial direction inside the outer core.The latter is very crucial because will give us a full picture and will help us to finalize the figure at the ICB so as there is compatibility between the percentages of [8] and field intensity.
Although this study could be terminated at this point as a short review paper, it is very instructive to continue the presentation and deepen in some alternative views of modeling Earth's interior.In section 7 we start with the standard IGRF model which, as we shall show below, is not capable of accurately representing the true singularity inside the outer core, thus we have to resort to other models as well.Within this context, we further test the following models: (i) Classical inversely-cubic model based on the magnetic dipole moment (equations ( 10) to (13)).(ii) Novel inversely-cubic model based on the RMS value of field on Earth's surface (equations ( 27) and (28a), (28b)).
(iii) Tiny bar magnet model.
Regarding all the above four models, when the observation point P moves to infinity the distance r increases thus the field decays.In contrast, different behavior occurs when the point P moves toward the Earth's centre.More precisely, the first two models (inversely-cubics) become singular only at Earth's centre whereas the last two become singular at the poles of the magnet.

Testing the IGRF-13 model
Although the IGRF-13 is a multipole expansion based on 13 harmonics in its current version [18], figure 4 shows that the outer space is dominated by the inverse-cubic law ( = n 3).The percentages shown in the bar diagram have been calculated as the ratio of the harmonic W n of equation (23) over the total field energy in the infinite outer space (atmosphere and interplanetary space).
Furthermore, figure 5 refers to the mantle and the liquid outer core and shows that as we move from the surface toward the Earth's interior, progressively the higher harmonics appear.Each sub-figure refers to a specific point P of the observer, while the total energy shown in each sub-title is between the said point P and Earth's surface.Note that the percentage refers to the contribution of each harmonic in equation (20).
Finally, figure 6 refers to the energy spectra of the solid inner core.To avoid extreme singularities, the closest to the Earth's center one-quarter of its radius (i.e., the part ) has been excluded.Again, in the inner core results are illustrated for the region   r 304 km 1212 km, or equivalently for   r R 0.0477 0.1909.

/
It is clearly shown that if all the thirteen harmonics are used, the calculated energy becomes unrealistically tremendous and only the highest harmonics appear.Now, we shall compare the IGRF-13 model with the inverse-cubic model (equation ( 27)) also including a major part of the core.The latter is throughout based on the reference value B RMS E , (for the year 2020 equal to 43,700 nT).An overall variation of the total magnetic energy trapped in a hollow sphere of variable inner radius r and fixed outer Earth's radius R E is shown in figure 7 (double logarithmic scale).In the same figure the ) are marked vertically to the x-axis.In this big picture one may observe that the difference between the two models inside the mantle ( ) is minimal but inside the core it dramatically increases.The theoretical reason that the IGRF model leads to values that become larger and larger as the inner radius approaches Earth's center has been fully explained in section 5. 4.
In general, the inverse-cubic-law leads to reasonable and finite figures.In more detail, the curve which is based on the (~r 1 3  / ) law intersects the Verhoogen's value (∼7.05×10 21J) at the position = ŕ R 0.0485 (not shown), while for a still smaller radius such as = ´= ŕ R R 0.1 0.01909 inner_core E the same model gives the much larger value of (∼1.2×10 23 J).Interestingly, at a still smaller radius of magnitude = ŕ R 0.0031 E (i.e., outside a sphere of radius equal to 1.65% of the inner core's radius, or 0.31% of Earth's radius) the r 1 3 ( ) / -model gives exactly the value of the total Earth's energy at the time of formation (∼2.5×10 25 J).
In contrast, the full IGRF-13 model intersects the Verhoogen's value (∼7.05×10 21J) at = r R 0.3646 E which is a point near the middle of the outer core, as shown in figure 7.In other words, between the radius = R R 0.3646 limit E and R E (i.e., within the outer 36.5% of the Earth's radius) the trapped magnetic energy is  calculated to be equal to the totality of (∼7×10 21 J) according to Verhoogen (1980).We recall that (as previously mentioned) the same energy of (∼7×10 21 J) had been calculated between = R R 0.0485 E min (= 309 km) and R E (= 6371.2 km) in the inverse-cubic model, which sounds to be reasonable.
In an alternative way, figure 8 presents the RMS as well as the extreme (min-max) values of the magnetic field on the same spherical surfaces.Again, the illustrated curve labeled as r 1 3 / has been constructed by implementing equation (27).For a typical variation of the RMS value of the field B along a certain meridian plane in terms of the latitude, the reader may consult [26].
Again, in figure 8 we present the field B in Earth's interior as a function of the normalized radius r R E / for the same two models.The symbol ICB stands for the 'Inner-Outer Core Boundary' which appears at = r RE 0.1909.

IOCB ( ) /
With respect to the full IGRF model, while the maximum value B max follows a smooth curve, the same is not the case for the minimum value B .
min Nevertheless, it was verified that the statistically calculated RMS value (according to equation (5)) coincides with the analytical one using equation (13a).
Regarding field values (in mT) reported in established literature (see section 2.3), one may observe that:  ( From the above discussion it comes out that the IGRF-13 model is not applicable in the major part of the outer core and particularly inside the inner core, as it leads to extremely high numerical values.The latter is because the radius r (distance from Earth's centre) appears in the denominator raised to a high power (up to the r 1 27  / ) thus infinite values are produced when we approach it.Nevertheless, if specific data were available, one could somewhat experiment with the crowd and the order of the lowest most suitable harmonics.It will be later shown (see section 11.2.2) that the first three to four harmonics perform better than all the thirteen ones which lead to figures 7 and 8.

Testing the dipole moment model 8.1. The original model
This model is a special case of the IGRF model, in the sense that is related to its first harmonic, and refers to equation ( 10)-( 13).We recall that equation (13) works well for the upper bound = ¥ r 2 ( ) but it requires a lower bound that must be higher than zero ( > r 0 1 ). Applying equation (13) between the unknown interior position r 1 and the Earth's surface, = r R ,

( ) 
For both the above cases, the distribution of the trapped energy in terms of the distance r is very steep as shown in figure 9, where one may observe that the difference between the two estimations of the total energy (i.e., D = ´-Ẃ 7 10 9 10 J total 21 20 ) is about 87% of the largest value ( 7 10 21 J) and occupies a short part along the horizontal r-axis.
Furthermore, if -for example-we adopt the estimation reported by Verhoogen [25] ( = Ẃ 7 10 dipole 21 J), the same dipole law depicts that almost the totality of this energy, i.e.
6.8896 10 21 J (98.4%) is contained in the region   r 305.7km 1216 km (part of the inner core).Similarly, if we adopt the estimation reported by Bullard [24] ( = Ẃ 9 10 dipole 20 J), the same law depicts that almost the totality of this energy, i.e. 7.8747 10 20 J (87.5%) is contained in the region   r 606.0 km 1216km (also part of the inner core).This contrasts with the report by Glatzmaier and Roberts [8] where it is claimed that no more than 10% of the total energy is included within the inner core.The reason is obviously because the inverse-cubic law considers the source at the Earth's centre while it is probably distributed within the liquid inner core where the currents of melted Fe and Ni take place thus producing magnetic field.Unfortunately, here the role of inner and outer core has been inversed and this is due to the fictitious singularity at Earth's centre.
Closely related, figure 10 shows the magnetic field (RMS value) where unrealistic values appear in the inner core.
In conclusion, the dipole model ('as is') suffers from the unrealistic inherent consideration that a singularity occurs at the Earth's centre, a fact that pushes the curve toward the aforementioned centre.Thus, this model is not applicable to the outer and particularly to the inner core.

The piecewise constant dipole model At this point a new Theorem regarding the RMS value of the dipole model is formulated and proved in Appendix
A. Although in this subsection the theorem is applied exactly to the ICB, it is generally applicable to every point inside the Earth.The limit case of Earth's surface is given in Appendix A, as numerical example.
In this context, based on equation (12b) we determine the RMS field intensity at the ICB by: Then we assume that the entire inner core (   r R 0 ICB ) is dominated by uniform field of intensity B , ICB while outside the ICB ( > r R ICB ) the usual dipole law occurs.Under these conditions it is very easy to show that the energy trapped into the inner core is exactly equal to the energy outside the ICB and both are given by: which is smaller than 9 10 20 J [24].Interestingly, the RMS value of field intensity in the outer core is given (by virtue of equations ( 3) and (13a)) by: After numerical substitution in equation ( 42) we obtain B RMS CD ( ) =1.3 mT, which is very small compared to both the well-known values of 2.5 mT [7] and 4.0 mT [28,31].
Therefore, although the piecewise constant dipole (CD) model may cut the singularity at Earth's centre, it leads to smaller energy content than [24] and also to a 50:50 distribution instead of the accepted ratio, inner-toouter core equal to 10:100 (claimed in [8]).

A comparison of the three models into the Earth's mantle
Let us now focus on Earth's mantle In figure 11 we compare all the three methods in a single diagram, where (i) the IGRF-13 model results in 5.8821 10 18 J, (ii) the inverse cubic model according to equation (27) calculates 4.2021 10 18 J, while (iii) the dipole according to equation (13) gives 3.9098 10 18 J.Also, in figure 12 we present the RMS values of the associated field intensity.These two figures should be compared with figures 5(a), (b), (c) in which one may observe the breakdown of the involved harmonics of the IGRF-13 model.Therefore, the difference between the accurate IGRF-13 model and the inverse-cubic models is due to the fact that higher harmonics somewhat contribute.

The weakness of the bar magnet model
Due to the fact that the model of the magnetic dipole (see section 8) has the inherent shortcoming that it becomes singular at the Earth's centre, while the singularity probably does not exist (the source of magnetic field is attributed to the liquid outer core), we test several models of the tiny bar magnet and then we continue with other models.Below we present four models, as follows: • Model-1: Tiny bar magnet with poles on Earth's surface.
• Model-2: Tiny bar magnet with poles at critical distance according to equation (9a).
• Model-3: Magnetic dipole with poles in the middle of the liquid outer core.J in the Earth's atmosphere [26].
In more detail, the difference between the above negative value W bm 12 and the total amount of = Ẃ 7 10

( )
By virtue of equation (7), the radius carrying the magnetic substance will be: / which means that the self-energy could be accomplished through a sphere of small radius R , i compared to that of the Earth's radius R , E which would be centered at each pole.This model gives an almost vanishing field at Earth's centre: ( ) The shortcoming of the above model (Model-1) is that it cannot accurately represent the field (i) neither at the two magnetic poles where it becomes infinite (ii) nor at the equator where it gives a rather short value (about one-third of the anticipated 30,000 nT).In addition, it leads to a vanishing field at Earth's centre which is not reasonable.Overall, this model must be completely rejected.
Therefore, the results of this model (Model-2) are very close to the accepted values of 60,000 nT and 30,000 nT at the poles and the equator, respectively.Not only that, but now the updated interaction energy highly increases to 7.0008 10 21 -J (by construction, its absolute value equals to the total energy).To make this point clear, for the critical distance d min the interaction energy is total Therefore, each self-energy equals to ~7 10 21 J, which -as said above-eventually gives the total magnetic energy of ~7 10 21 J.
Regarding the field at the Earth's centre, Model-2 predicts: which is a huge number, not reported in the literature so far, thus cannot be accepted.Remark: If the above calculations are repeated considering the total magnetic energy as that estimated by Bullard [24], i.e., equal to 9 10 20 J, similar negative conclusions will be drawn.Therefore, Model-3 somewhat deviates from the accepted values of 60,000 nT and 30,000 nT at the poles and the equator, respectively, and gives a field at Earth's centre which is 10 times smaller than what is expected.

Cylindrical bar magnet
An older study by Davis et al [6] shows that the energy stored in the external field of a right cylindrical bar magnet is proportional to the square of the intensity of magnetization M̅ and a non-dimensional geometrical factor A which is in turn a function of the ratios of geometric length to diameter (q) and of magnetic to geometric length (b).Under certain conditions an approximate expression for the field energy per unit volume of the magnetic material is obtained after regression analysis, which is applicable 'as is' for the CGS-system.Then, the total field energy E for cylindrical magnets are calculated merely multiplying E v by the total volume of the magnet, V .magnet To avoid unit conversion difficulties, the magnetization M̅ was converted to [emu/cm 3 ] multiplying by 10 3 , so the energy density E v was found in [erg cm −3 ] and the energy E in [erg], which was eventually converted into joules (1 erg = 10 −7 J).Using = Ḿ 7.7460 10 A .m 22 2 (to achieve identical results with the IGRF-13 model in the far field), table 11 shows how sensitive the numerical result for the calculated energy is.One may observe that the calculated total infinite energy ( 8 10 17 J) using the IGRF-13 model correspond to an aspect (slenderness) ratio approximately equal to = q 1.759.Also, the well-known values of 9 10 20 J (Bullard [24]) and 7 10 21 J (Verhoogen [25]), were found for = q 1979 and = q 15388, respectively.Note that, except of q=0.5 and 1, in all other cases the Earth's volume It is worthy to mention that using this model one could also consider the case that the position of the magnetic poles are not exactly on the Earth's surface but are located at s from the center of the magnet (with < s R E ) and on its axis.Then we have to deal with magnetic to geometric length ratio, b, which is mentioned above.The results of table 11 show that, for a prescribed magnetic moment M, the shape of the magnetic model has substantial influence on the energy around the magnet.If the magnet includes the center of the Earth and is close to a spherical shape, then the above theory deviates from reality.But in any case, by subtracting the energy beyond the earth surface, it may give us an estimation of the geomagnetic energy entrapped inside the Earth.

Comparison between three models
In this section we shall compare (i) the IGRF-13 model with (ii) the dipole model (equation ( 10)-( 13) based on the magnetic moment M) and (iii) the inversely-cubic model (equation ( 27) based on RMS value of Earth's surface).
Regarding Earth's exterior, the computations are shown in table 12 at the row labeled as 'Atmosphere'.In more detail, within Earth's atmosphere and interplanetary space, the IGRF-13 model (based on 13 harmonics) predicts 8.0165 10 17 J, the Dipole model gives 7.7332 10 17 J thus underestimates (error −3.53% with respect to IGRF-13 model), while the Inverse-cubic model leads to 8.2768 10 17 J thus it overestimates (error +3.25% with respect to the IGRF-13 model) the corresponding energy.
Concerning the liquid outer core (between ICB and CMB), the IGRF-13 model leads to an unrealistically large value ( 2 10 29 J) while the dipole and the inverse-cubic model give both the somewhat small magnitude of ~1.1 10 20 J.The latter value is ten times less than the magnitude of (10 21 J) reported from Landeau et al  [31] as well as Braginsky and Roberts [30] for the entire core.
Finally, regarding the inner core (inside ICB), none of the three models can approximate the energy because in all these cases powers of the radius r (i.e., r 3 up to r 27 ) are involved in the denominator of the expansion series for the field.
Interestingly, either of the three above models is adopted the conclusion is practically the same.More precisely, the sum of the energy in the mantle and the atmosphere (i.e., outside of the CMB) varies between ´ḿin 4.7210 10 , 5.0530 10 ( ) J and 6.7045 10 18 J, accordingly.Therefore, with reference the estimation of the total energy according to Verhoogen [25] ( 7 10 21 J) or Bullard [24] ( 9 10 20 J), respectively, the three models predict that the energy trapped in the exterior of the CMB is 0.06 0.75% or 0.10 0.47%.In other words, in all the three models the calculated percentage is not more than 1%, thus is according to the limits proposed by Glatzmaier and Roberts [8].
Nevertheless, the two approximate models are very close one another when referring to the calculation of the energy between ICB and Earth's surface (i.e., outside the ICB).In more detail, the dipole model predicts ( 1.1123 10 20 J) while the inverse-cubic model predicts ( 1.1905 10 20 J).The latter lead to a percentage of 12.4 13.2% with respect to [24] and 1.6 1.7% with respect to [25], of energy between ICB and Earth's surface which is out of question (extremely small) compared to the anticipated 89 90% according to Glatzmaier and Roberts [8].
In general, the inverse-cubic models within the Earth give an energy which depends only on the position r and the magnetic moment Μ.As these models are incapable of determining the total energy in the entire globe (due to the singularity at its centre), the percentages highly depend on the selected basis reference, i.e = Ẃ 9 10 total 20 J or 7 10 21 J.In all cases, a single dipole at the Earth's centre cannot give the desired limits proposed by Glatzmaier and Roberts [8], i.e. 10% in the solid inner core, 89% for the liquid outer core, and 1% for the mantle.From the other hand, multiple poles centres lead to huge magnitudes of energy.Therefore, the key-point is to properly simulate the liquid outer core which can be done by the numerical solution of the MHD dynamo PDEs (Navier-Stokes equations) [7][8][9][10][11].

Bridging the missing gaps 11.1. Position of the problem
In this section we discuss possible scenarios according to which the gap of the field intensity values between the ICB and CMB can be bridged.First, Buffett [7] proposed a core-averaged field equal to 2.5 mT and from a closely related official newsletter that focuses on the aforementioned paper, we also learn that this figure corresponds to a point inside Earth's core, 1,800 miles (approx.2897 km) underground (Sanders [27]).On the other hand, Landeau et al [31] report the RMS field value of about 0.42 mT on CMB and also claim that the value on ICB which is anticipated to be within the interval -30 50 [ ] mT.Actually, in section 8 (figure 12) we have validated the value of 0.42 mT (implementing the IGRF-13 model), which has been to be bridged with the aforementioned known value at the ICB.Moreover, Gillet et al [28] have previously reported the value of 4 mT within the liquid outer core.Based on these data we shall try to propose a reasonable distribution of the field strength between the (unknown state at) centre and the known state at the CMB.We recall that so far, we have seen that the dipole (CD) model fails to determine a proper averaged field value for the outer core (it predicts 1.3 mT, which is lower than either of 2.5 mT and 4.0 mT) and also it gives the value of only 0.26 mT at the CMB (lower than the anticipated 0.42 mT).
To comment on the aforementioned finding of 4 mT (or 2.5 mT) inside the outer core, we consult figure 12, where one may observe that exactly on the CMB the average (RMS) field is ∼0.42 mT, a finding based on the IGRF-13 model.Therefore, this figure is much smaller than the value of 2.5 mT, which in the original paper is characterized as core-averaged field (Buffett [7]) and in no way exactly on the CMB as Sanders [27] claims in layman's terms.Actually, in figure 8 one may observe that IGRF and inverse-cubic models (both erroneous) determine the value of (2.5 mT) at = r R 0.4542 E and = r R 0.2595 , E respectively.This means that in both cases the RMS-value of (2.5 mT) actually is inside the liquid outer core.
Our further investigation on the average value of the field inside the liquid outer core (after analytical and/or numerical Gaussian or Simpson integration) has shown that: (1) The inverse-cubic model leads to the value (∼1.3 mT), which is smaller than (2.5 mT).
(2) The IGRF model with its all 13 harmonics leads to the extremely high value of (5.4×10 34 mT).
(3) The IGRF model with its first 5 harmonics leads to the value (18.9 mT) and the calculated value at the ICB is 168 mT.
(4) The IGRF model with its first 4 harmonics leads to the value (8.5 mT) and the calculated value at the ICB is 68 mT.
(5) The IGRF model with its first 3 harmonics leads to the value (3.8 mT) and the calculated value at the ICB is 26 mT.
In other words, the value of (2.5 mT) reported by Buffett [7] is around the middle of the average values estimated by IGRF model using three and four harmonics.Furthermore, if instead of 2.5 mT we consider the averaged value of 4 mT reported by Gillet et al [28], the fifth harmonic somewhat contributes (i.e., should be > n 4).This issue is further discussed in section 11.2.11.2.The need for some hypotheses 11.2.1.A first approach As we have previously seen in section 11, the function B r ( ) does not follow either the multipole IGRF-13 or the inverse-cubic law measured from Earth's surface (B ,

RMS E
, see equation (27)).A possible way to determine it would be to carefully develop a MHD dynamo model using a structured spherical mesh which would allow us to determine the RMS value of B r ( ) at each radius r thus producing the aforementioned function B r .RMS ( ) Then it would be easy to calculate the desired overall RMS value in the liquid outer core and test whether it is actually equal to 4 mT.But since this task requires huge effort and might be the topic of a specialized full length original research paper, for our current needs we must resort to a much simpler solution.
On the other hand, if we adopt the receipt of Glatzmaier and Roberts [8] according to which equation ( 36) is valid, we obtain the following lower limit for the RMS field intensity in the outer core: It is worthy to mention that the abovementioned calculated lower bound of 3.1 mT is larger than the value of 2.5 mT proposed by Buffet [7] and lower than the usual value of 4 mT reported in [28,31].Therefore, we have adequate confidence to definitely assume that = B 4 mT.
Therefore, we can safely assume the limit case = B 6 mT inner core (RMS value), which however is much smaller than the values [ 30 50] mT at ICB (proposed by Glatzmaier and Roberts [8]).In other words, the limits of equations (35) and (36) regarding the field RMS intensities are in contradiction with the reported values [ 30 50] mT at the ICB of the same authors unless there is an abrupt discontinuity at ICB or a progressive decrease into the inner core, which is not very likely to occur.
The above case in which the value of 0.42 mT at the CMB is smoothly connected with the value of 6 mT at ICB is shown in figure 13.Clearly, even a simple quadratic polynomial might control the RMS field inside the outer core to become exactly equal to 4 mT.In addition, a cubic polynomial may achieve the same RMS value of 4 mT as well as a vanishing slope (zero derivative) exactly at the ICB thus offering the possibility that the field continues uniform till the Earth's centre (for a detail, see figure 14).Of course, the reality is expected to be quite different, and this is just to show one form out of the many possible patterns to obtain the accepted RMS value of 4 mT.At the same time, one may observe that if the transition from 0.42 mT to 6 mT was linear (blue line in figure 13) the resulting RMS field would be about 2.8 mT, which is smaller than the expected 4 mT.Moreover, the convex curves (for their definition see section 11. It is clarified that the quantity B r ( ) in equation (48) is the RMS field intensity on the surface of a sphere of radius r.In other words, the global RMS value of the field is actually a volume integral, which can be calculated in two successive steps.First, we calculate the RMS value of the field on many spherical surfaces of radius r and then we use the latter values into a univariate integral with limits [R R , ICB CMB ] to calculate the desired RMS field value.Again, in the hypothetical case of a uniform field of intensity 6 mT inside the inner core, the transition shown in figure 14 shows a continuity at ICB (be construction) and a discontinuity of the slope at CMB.

A second approach
In the sequence we shall make assumptions about the reported interval of [ 30 50] mT at ICB, which for the sake of simplicity is assumed to be uniformly extended from ICB to the Earth's centre.For the sake of completeness, we extend the left bound of this interval from 30  (to fulfil the criterion that the magnetic energy of the inner core is 10% of the energy in the outer core), equals the usual RMS value of 4 mT which is repeated in the literature (e.g., [28,31]).Consequently, if the RMS value of 4 mT within the outer core is taken for granted, for any RMS value at the ICB
A general remark is that the value of 2.5 mT reported by Buffet [7] is not included in table 13 because, according to equation (49), the maximum RMS value at ICB fulfilling the energy-ratio criterion of 10% is only 3.8 mT.Another remark is that if we eventually consider the value of 4 mT (reported by Gillet et al [28]) as a reference for the outer core, table 13 shows that the calculated RMS value according to the inverse-power law (for » n 3.83) corresponds to 23.8 mT at ICB (i.e., smaller than the lower bound of the interval 30 50 [ ] mT), where the energy-ratio is strongly violated becoming l ¢ = 1.6 0.1.This value is the closest one to the desired limit of 26.6 mT, which is associated to energy ratio 0.1 (10%).Of course, the inverse-power law described by equation ( 50) is tentative to allow us extending our discussion but in any case, the radial variation of the field for all the three values at ICB (30, 40 and 50 mT) are illustrated by convex curves in figure 17.In the latter case the slope discontinuity moves to the ICB (instead of the discontinuity at the CMB shown in figure 14).
A parametric study for a larger variety of RMS values of field intensity at the ICB and the above-mentioned five assumptions is shown in table 13.One may observe that the desired averaged value of 4 mT in the liquid outer core occurs when = B 23.8mT ICB (inverse-power assumption) in conjunction with l = 1.6 0.1,  which however is outside the interval 30 50 mT [ ] suggested by [8].This conclusion is not based on any special assumption except that the function B r ( ) is smooth and monotonic between the two ends.The last column in  (∼7.05×10 21J) corresponds to the total magnetic energy.In addition, the aforementioned amount is split in two parts, the former outside the core (W 1 = 5×10 18 J to 5×10 19 J) and the latter inside the core (W 2 = 7×10 21 J).
Regarding the core, it contains poloidal (with radial B r component) and toroidal f B (donut-shaped) fields.Although the latter are not detectible at the Earth's surface, nevertheless they play an important role in the production of the magnetic field.The analysis of the core is facilitated considering balance between Coriolis and Lorentz forces as well as tidal friction and nutation.In his report, Verhoogen (1980) [25, p. 73] estimates the grand maximum magnetic field at about 30 mT (300 G).
In an older report, Bullard (1949) [24] discusses the inverse cube law to the field observed at the surface and derives a possible value of 4 mT (40 G) for the toroidal field while the total energy is estimated to about 9×10 20 J. Later works suggest an internal magnetic field of 1 4mT ( 10 40G ) (Aubert et al [56]; Christensen and Aubert [57]; Starchenko and Jones [58]).
Previously, it has been reported that the total power released due to radioactive heating and matter crystallization is 10 ± 4 TW, which is sufficient for magnetic generation requiring 1-5 TW (see, [53]).In a later report, the core-mantle boundary (CMB) heat flux is estimated at 12 ± 5 TW and is claimed (a smaller than previously amount) to be sufficient to drive a dynamo dissipating 0.1-3.5 TW at present (see, [53]).Below we shall try to validate the latter figure ( 0.1 3.5 TW) thus to check whether the IGRF or the inverse-cubic model is closer to it.
Actually, according to equation (13a) the total magnetic energy is proportional to the square of the dipole moment.Similarly, Davis et al [6]

( ) ( )
Since the above work is spent within 5 years, the corresponding power in TW will be: (a limit ensured only in the inverse-cubic model), it is easily concluded that the total magnetic power of ∼0.18 TW associated to both the interior and exterior of the Earth: (1) Is higher than the lower bound (0.1 TW) but 20 times smaller than the upper bound (3.5 TW) which have been reported by [53]).
(2) Is slightly greater than the amount of 0.14 TW proposed by Bullard [24].

A second test and final question
In Landeau et al [31] it is mentioned that decadal to secular fluctuations of the geomagnetic signal suggest a largescale velocity of ~´--3 10 ms 4 1 below the core surface.Assuming that this figure is representative of the flow in the bulk of the outer core, yields a kinetic energy of 8 10 J 16 which is about 10 4 times smaller than the magnetic energy.This is an implicit assumption that the total energy is of the order ¸8 9 10 J, 20 ( ) which practically is in favour of the report by Bullard [24].By the way, a question to the readers could be as follows.What percentage of the total Earth's kinetic energy is the total field magnetic energy?
Answer: The total Earth's magnetic field energy is negligible compared to its total kinetic energy.Actually, considering 365 days per year (in which the Earth completes a period around the Sun at radius 1 AU, i.e., approximately 149.6 million kilometers) as well as the 24 h per day (in which the Earth, of radius 6371.2 km, completes a period around its axis), one may easily find that the corresponding kinetic energies are equal to = É 2.67 10 13.A first feasibility study regarding the energy absorption from Earth's interior The continuous feeding of the magnetic field by the Earth's interior is necessary to sustain the magnetic field over long periods of time.When we talk about losses in the context of the Earth's magnetic field, we are referring to the processes through which the magnetic field's energy is dissipated or weakened in the atmosphere and of the field, which somewhat differs from the arithmetic mean value (see Provatidis [26]), but there is a large range between minimum and maximum values on it.Since the published results are usually based on energy issues, it is hypothesized that should refer to RMS values.This study was not able to propose a definite value for the total Earth's magnetic field energy, 9 10 20 J or 7 10 21 J, but several points in this study are in favour of the estimation that the reality is probably in-between.
Overall, a lot of practical issues have been elucidated mostly regarding the magnetic field, and the major topic has been the percentage of the energy content within the inner core compared with the one within the outer core (less than 10% according to [8] or not).
Some of the findings are as follows: (1) Regarding the magnetic Earth's total field and the relevant energy inside the mantle and the surrounding atmosphere, the golden standard is the so-called International Geomagnetic Reference Field (IGRF-13) model.
(2) Using the abovementioned IGRF-13, the total energy outside the core (inner and outer, i.e., outside the CMB) has been established at (~6.7 10 J
(3) The IGRF-13 standard was extensively compared with the classical dipole well known from in physics textbooks as well as a novel inverse-cubic model based on the RMS (homogenized) field on Earth's surface.The last two models differ from the IGRF-13 by 3.5% in the atmosphere, while they underestimate the energy in the mantle by about 28%.
(4) The difference becomes still higher when all the three models are applied to the outer core and tremendously larger when implemented in the inner core.While the total energy outside the core (inner and outer) has been established at (~6.7 10 J

18
), the same is not the case for the core itself.More precisely, the IGRF-13 model becomes problematic when applied (with all its 13 harmonics) to the core merely because it leads to tremendous figures, thus an alternative approach is needed.All models tested were found to be singular near the Earth's centre thus not applicable in this region.
(5) In this context, a novel inverse-power model based on the RMS field at the CMB was found to represent the radial variation of the field in the outer core in a reasonable way.
What could be the next step?If the existing computational models such as [8] and all relevant results were available, we could proceed to a post-processing to determine the unknown distribution of the RMS field intensity B r ( ) in terms of the radius r.We recognize that the primary concern of those pioneering investigators was to develop their multi-physics model from scratch and probably to perform a sort of sensitivity analysis regarding the most influencing parameters and their dependencies, a difficult task which does not allow focusing on details.The imposition of the boundary conditions in a Computational Fluid Dynamics (CFD) model is of major importance.As far as one can understand from reading, Neumann conditions of heat flux have been imposed while today it would be possible to directly impose the particular value of the field intensity B x y z , , ( ) at every nodal point on the boundary (usually the CMB), simply implementing a computer software such as [41].Not only that, but it becomes imperative to generate a structured computational mesh which will follow spherical surfaces of a specified radial step Dr.Then, taking the nodal points or/and the centroids on the spherical surface (face of these 'finite elements' or 'finite volumes') it would possible to perform post-processing and calculate the RMS value of the field intensity, B r , ( ) at each radius r within the interval through the formula: This information is still missing and could be the topic of a cooperative research program implementing several independent computer codes.After the function B r ( ) has been determined, we can proceed applying equation (48).
Therefore, since the above information has not been documented in any published report [7][8][9][10][11], the author had to rely on older resources such as [24] and [25] which estimate the totality of Earth's magnetic field energy between 9 10 J 20 and 7 10 J. 21 Moreover, a second resource for our study was the breakdown proposed in [8], which suggests that the energy in the inner core is (at maximum) the 10% of the energy within that in the liquid outer core (see equation (35)).Moreover, the energy in the exterior of the outer core (~6.7 10 J 18 thus small compared to either of the aforementioned 9 10 J 20 or 7 10 J 21 ), is at maximum 1% of the total energy content.
But again, another theoretical problem arises.What about the distribution of the field inside the inner core?Based on current scientific understanding, it is believed that the magnetic intensity inside the Earth's inner core does not vary significantly from point to point.The inner core is a relatively homogeneous solid metallic sphere, primarily composed of iron and nickel.However, it is important to note that the inner core's magnetic field is still subject to ongoing scientific research, and the available data is limited.The inner core's magnetic field is primarily generated by the motion of the liquid outer core, which surrounds it.The complex dynamics of the outer core can influence the magnetic field generation, and this, in turn, may have some impact on the magnetic intensity within the inner core.
While there may be small variations in the inner core's magnetic field, the available evidence suggests that any spatial variations from point to point within the inner core are likely to be relatively minor.The inner core is generally considered to exhibit a stable magnetic field with consistent intensity.However, it is worth noting that obtaining direct measurements from the inner core is extremely challenging due to its inaccessibility.Most of our knowledge about the inner core's magnetic field comes from indirect observations and modeling techniques.As research continues and more data becomes available, our understanding of the inner core's magnetic field variations may evolve.
It is worthy to mention that the assumption of a uniform magnetic field in the inner core, in conjunction with the reported values of  mT at the ICB and 4 mT in the liquid outer core, leads to energy breakdown which does not fill the abovementioned criterion 1:10 proposed in [8] (in general leads to higher energy in the inner core).This is for reasons we can only hypothesize about.For example, there may be a jump of the field at the ICB so that the uniform value inside the inner core is smaller than what the ICB dictates.Another reason may be that the calculated value of 4 mT is small for several technical unknown reasons.But, again, it is of major importance to replicate older works regarding the computational models of the Earth's interior with imposing and/or testing the IGRF-13 field values which all of us assume and expect to be valid on the CMB as well.
To bridge the gap between the value 0.42 mT at the CMB and the stated values between 30 and 50 mT at the ICB, several scenarios (such as linear interpolant, inverse-power, exponential, logarithmic, and polynomial) were offered.The difficulty to be in favour of a certain scenario is due to the fact that we have to determine such a supposedly smooth function B r ( ) that its RMS value according to equations (48) or (53) is near 4 mT while the energy ratios in the cores fulfils the criterion l =  U U 0.1 .
inner core outer core ( ) / While the abovementioned ratio 1:10, described by equation (35), depicts that the ICB has a field of about 6 mT, this figure is quite smaller than the reported region between 30 and 50 mT.In brief, the value of 6 mT is easily bridged by a concave curve which fulfils the condition 1:10 by equation (35), while the values 30 to 50 mT can be also bridged but with a convex curve which does not fulfil the condition 1:10 by equation (35).
If the primary concern of a reader is to calculate the total energy of Earth's magnetic field, he/she has the opportunity to deal with the energy associated to a bar magnet.From the didactic point of view, this is a fourth model (in addition to the dipole, inverse-cubic and the IGRF-13 standard) of a tiny bar magnet which is extended from the North to the South Pole (more accurately tilted at an angle of about 11°with respect to Earth's rotational axis).Actually, in such a virtual case, the 'interaction' energy is about »-Ẃ 3 10 , 12 17 of which the absolute value is smaller even than the total energy in the infinite space excluding the Earth (∼8×10 17 J), not even considering Earth's interior.Therefore, we have to consider the self-energies due to the poles.Another shortcoming of this idea is that the field at the poles (not at the center) would become enormous high, which is not a real fact.Having said this, an improvement to this idea is to introduce the concept of a right cylindrical bar magnet (i.e., long as the previous one but of a certain diameter D) in which the stored energy is proportional to the square of the magnetization and a non-dimensional geometric factor (aspect ratio) = q d D / (see, Davis et al [6]).For magnetic poles separated by distance = d R 2 , E for = q 1.8 we obtain the desired magnitude of (∼8×10 17 J), while for = q 15650 we obtain (∼7.05×10 21J).The latter model refers to a long and adequately thin cylindrical bar magnet, but it includes the singularities involved by the magnetic poles and also leads to a very weak field at Earth's centre.
Final Remark: The reported RMS value of 4 mT, is a crucial figure to characterize the averaged state of the outer core.With respect to other RMS values in physics (such as the voltage of alternating current), the difference is that in the latter case the integral is taken with respect to time while here it is taken with respect to the ).But similarly to the alternating current, the harmonic dependence of the field with respect to the latitude q (in the Central Dipole (CD) model) leads to the fact that the RMS value equals to the peak value divided by 2 (see, [26]).We should pay attention that in the IGRF model the ratio max-to-min is much higher.
To give an example, in the particular case of the CMB, the IGRF-based RMS value of the field intensity is 0.42mT while for the Greenwich meridian plane the minimum and maximum values were found to be 0.16mT and 0.72mT (see, [26]), respectively, thus the ratio peak-to-RMS value equals to 1.75 (i.e., higher than the usual » 2 1.41).Furthermore, as shown in figure 8, the ratio becomes still larger.
In any case, the maximum value of the field is larger than the RMS value, so we must take care whether the Again, this study has considered that the reported value of 4 mT has been considered as the RMS value for the entire outer core.

Conclusions
From the several models developed and compared toward the estimation of field intensity and energy content inside the Earth, it was concluded that any judgment requires at least a working assumption.In the lack of consensus, the Maxwell's Second Law (divergence Gauss's law of magnetism) was considered in conjunction with a uniform magnetization thus leading to a uniform magnetic field inside the inner core.Then, assuming no field jump between the ICB and the inner core, it was shown that the energy content of the latter is fully determined by the total field at the ICB.Moreover, since the root square mean (RMS) value of the total field intensity inside the liquid outer core has been previously estimated at 4 mT it was possible to check that the usually stated field between 30 and 50 mT at the ICB leads to such a large energy content within the inner core that is much more than the accepted value of 10% of that within the outer core.This finding is in contrast at least to one of the established pioneering MHD dynamo computational models.In the lack of data, it was assumed that the total field about 0.42 mT at the CMB could be easily bridged with a supposed value of 6 mT at the ICB through a concave curve thus ensuring that the energy of the inner core is no more than 10% of that within the outer core.Alternatively, large values such as 30 and 50 mT at the ICB could also be bridged with the 0.42 mT at the CMB through a convex curve, however then the condition of 10% is no more fulfilled.The findings of this work show that a detailed post-processing of the computational results in the MHD dynamo model is imperative to establish the radial variation of the total RMS magnetic field at least between the ICB and CMB.Moreover, the variation of the total field inside the inner core is of great importance as well.
Comparing (A-3) with equation (A-2) we obtain the desired equality:

Figure 1 .
Figure 1.The three main layers of Earth's Interior.

Figure 2 .
Figure 2. Bar magnet and resulting field.
Based on equation (12a) in conjunction with equation (3), after performing the definite integral ò

infinite 17 Interestingly
, as for the year 2020, the ratio of the first harmonic = Ẃ 7.6588 10 J 117 (see equation (23)) over the entire energy = Ẃ 8.2315 10 J E,inf17 mentioned in section 5.4 equals to 0.9304.The latter value is exactly the figure of 93% which Alken et al[18, p. 20]  have mentioned as the contribution from the dipole terms ( = n 1: g g h positions of the CMB ( = r 3486 km with = r R 0.5471 E / ) and the ICB ( = r 1216 km with = r R 0.1909 E /

Figure 4 .
Figure 4. Spectrum of total energy in the infinite outer space beyond Earth's surface (year 2020).

Figure 5 .
Figure 5. Energy spectra between Earth's surface and a point P either into either the mantle or the outer core for several ratios r R : E / (a) middle of mantle, (b) one-quarter the thickness of mantle measured from CMB, (c) exactly on CMB, (d) one-quarter the thickness of outer core measured from CMB, (e) two-quarters of the outer core measured from CMB, (f) three-quarters of the outer core measured from CMB (year 2020).

Figure 6 .
Figure 6.Energy spectra between Earth's surface and a point in the inner core for several ratios r R : E / (a) inner/outer interface (ICB), (b) three-quarters, (c) two-quarters, (d) one-quarter of the radius = R 1216 inner

Figure 7 .
Figure 7.Total field energy between Earth's surface and an internal spherical surface of radius r.

Figure 8 .
Figure 8. Geomagnetic field B (mT) between spherical surface of radius r and Earth's surface.

Figure 11 .
Figure 11.Magnetic field energy (in J) trapped inside Earth's mantle using three models.

in 21 JẂ 8 10 infinite 17 JẂ 7 10 total 21 J
(to which the outer amount of = is added) is covered by the self-energies associated to the two poles.Therefore, considering the estimation of = by Verhoogen[25], according to equation(5)
algebraic sum W f bm leads to + m p ,

9. 1 . 3 ./
Model-3: Magnetic poles separated by distance = d 4702 km As a last test for the tiny bar magnet, we choose the distance d which separates the two poles to be at the middle between the ICB and the CMB, i.e. at = forcing the singularities to occur at their actual position where loops of electric currents take place.Since now the interaction energy becomes = thus the radii of the spheres carrying the 'magnetic substance' are = = are entirely cited within the liquid outer core.Now, the superposition of the two poles according to figure3gives:Regarding the field at the Earth's centre, Model-3 predicts:

( 35 )
provides the following upper limit for the field intensity in the inner core: 2.2) lead to still smaller values, out of question.Note that for each of the abovementioned curves B r ( ) in the interval R R , ,

Figure 13 .
Figure 13.Hypothetical interpolation for bridging the RMS values of 6 mT (at ICB) and 0.42 mT (CMB).

Figure 16 .( 4 ) 4 ( 5 )
Figure 16.Numerical results for the different approximation models regarding the radial distribution of the RMS field intensity in the outer core.

5
Considering the above five admissible functions B r , RMS values are shown in figure16with the corresponding coloured dashed lines, whence one may observe a great dispersion of the results.

Figure 17 .
Figure 17.Bridging the ICB ( 30 50mT) with the CMB (0.42 mT) considering the inverse-power law in the liquid outer core.

Table 1 .
Reported values in literature.

Table 2 .
Evolution of magnetic moment and associated infinite energy.

Table 3 .
Geomagnetic field on Earth's surface and Energy breakdown outside the core-mantle boundary (CMB) using the IGRF-13 standard.

Table 4 .
Evolution of the trapped magnetic energy in the outer half of Earth's mantle.

Table 6 .
Extreme and RMS field intensity values at the CMB.
YearGeomagnetic field B (in [nT]) (19)a specific epoch between 2000 and 2020, the numerical value of W mantle can be selected from the sixth column of table 3, thus the RMS value for the entire mantle is calculated through equation(19)at about Regarding the mantle again, the minimum RMS value of field strength occurs on Earth's surface (43,700 nT = 0.044 mT = 0.44 G) while the maximum occurs on the CMB (0.42 mT = 4.2 G), an issue which was previously discussed in section 5.3.4 and will be further discussed in section 11.

table 8 .
Actually, we are looking for a representative root mean square (RMS) value B rms of the magnetic field which is associated to the above-mentioned total field energy of ] (for definitions see equations (3) and (19)).Since, by definition, the product of the aforementioned density u[J/m 3 ] by the volume V E [m 3 ] equals to the energy W total in [J]The difference of table8from table 1 regarding the averaged field is that the former (table 8) uses original results based on equation (19) while the latter (table

Table 8 .
Energy density and Averaged field calculated for the whole Earth.

)
The maximum value of 30 mT (300 G) is reached by the inverse-cubic inside the inner core at abour

Table 11 .
Dependence of the infinite energy on the slenderness factor q

Table 12 .
Breakdown of field energy in atmosphere and Earth's layers, as calculated by three models.
predicts that the associated lower bounds for the RMS field B outer core will lie between 4 mT (for the left bound of interval 6 50 [ ] ) and 33.29 mT (for the right bound of the same interval), respectively.In other words, the smallest value of them (i.e.,

Table 13 .
Calculated RMS field intensity in the outer core.
Hypothetical interpolation in the outer Core (l > 0.1) rms(mT)RMS value of field intensity in the outer core based on equation (49), have proposed a similar expression for a cylindrical bar magnet, i.e., (in A.m 2 ) is the magnetic moment, and a is a proper coefficient that has to be determined.Setting the dipole moment according to table 2 (for the years 2015 and 2020 equal to Numerical application: For example, if the point P is taken on Earth's surface, thus for = Ḿ 7.6460 10 22 Am 2 we have