Electromagnetic Induction within the Earth's Crust of Variable Conductivity (Part I)

Abstract

In this paper the earth's crust is considered to be consisted from m's concentric shells (including non-conducting shells) each of which is of uniform nature as to the electromagnetic state. This earth's crust is enclosed by a non-conducting medium, which substantially contains our atmosphere. The shells are bounded at the radii ξ_??_α (_??_=1, 2, … m; α being the radius of the earth).
It is assumed that the magnetic potential is expressed by spherical harmonic series as is given by (1) and (2) in the Japanese text. From the boundary conditions we obtain the relations (8), (9) and (10).
For the numerical calculation it is very troublesome to follow the abóve expressions. Now we wish to adopt the series of rather rapid convergency. For this purpose we must discuss the expression Rm. at first. For the small value of |kr| we use the formula (13) and for large value of |kr| we use (14).
The numerical values of |kr| for various xμ and modes of variations are tabulated in Tables 1 and 2, showing that |kr| is very large for the variation of rapid nature and for large value of xμ and vice versa.
When |kr| is very small the corresponding variation is of very long duration which shows us the lacking of suitable data to treat such a long periodic variation, hence we will confine ourselves to the cases of large value of |kr| only.
In the periodic variation we will adopt the relations (18) for the calculation. If we consider rather rapid variation the mathematical expression becomes very simple as we see from (22).
At any rate at the outermost boundary we have the relation (26). If |kr| is larger than 100 or for the variation of period shorter than about 1 hour, this expression is reduced to (27). In this case the ratio of the vertical component of the magnetic force due to the internal origin to that due to the external origin becomes (28), showing that the vertical component due to the internal origin is reversed its direction to that due to the external origin and their absolute values are generally of the same magnitude. In other words, in rather rapid periodic variation the record of the vertical force does not show any appreciable variation. The similar record is frequently obtained for the so-called Dellinger effect-this point will be discussed in the future paper.
If |kr| is not so large we must start from (26) and S₁⁽¹⁾ is to be transformed by (30). Here γ_??_is calculated as (31). This expression shows an important conclusion:
If there is at least a layer of sufficient thickness in the earth's crust the electromagnetic induction does not practically penetrate through this layer. This nature depends upon the period and the product of conductivity and permeability. The criterion for these quantities are expressed by the relation (32).
On the contrary, if we consider the aperiodic variation we can not find any convenient relation for the penetration of the electromagnetic induction and hence we must follow very troublesome calculation for the practical problem of the aperiodic variation.