LOSS OF THE EARTH'S MAGNETIC FIELD

DURING INVERSION

N. L. Vishnevskaya and L. A. Zashchinskii UDC 550.383

Nothing is known about the causes and mechanism of inversion (i. e., sign change in the Ear th ' s mag- netic field). Inversion plays an important par t in pract ica l uses of the theory of paleomagnetism, and the lack of a sa t i s fac tory descr ipt ion of the causes and course of inversion makes it difficult to advance a theory and make good use of a paleomagnetie dating scale . Also, the explanation of inversion must c lear ly be closely related to the general theory of the magnetic field of the Earth, and the origin of this field. At the present t ime, zones of r e v e r s e magnetization in volcanic rocks and sediments are explained in t e rms of the invers ion of the principal magnetic field of the planet [1, 2]. This explanation is applicable only if one assumes a se l f -exci ted t e r r e s t r i a l dynamo, but some other hypotheses [3] completely rule it out, which gives considerable in teres t to experimental and theoret ical discussion of inversion mechanisms.

It is impossible to construct hypotheses about the causes of inversion without a sa t i s fac tory descr ip - tion of the general course, but the available l i t e ra ture ca r r i e s only the crudest of schemes, in which the variat ion in the field s trength at any point of the surface is represented as in Fig. 1. A less crude approx- imation to the real situation may be the l inear variat ion during inversion as in Fig. 2. However, a scheme with a period of r e s t (section mn on the t axis, Fig. 3) is m o r e compatible with the hypotheses of Frenkel ' and E l sasse r , and other hypotheses of the t e r r e s t r i a l dynamo type [4]; the field strength on average is zero during this t ime. This period is related to the concept that there is d i sorder in the currents within the Earth, and the order ing of the t e r r e s t r i a l currents f rom the zero state takes place fairly slowly [5].

It is possible to pe r fo rm a mathemat ical tes t of the hypothesis of this quiescent period; our calcula- tion shows that the following two assumptions are incompatible: 1) there is a quiescent period during in- vers ion; 2) the magnetic field during the inversion remains analogous to a dipole field. The calculation consis ts in applying some concepts and consequences of the theory of charac te r i s t i c surfaces to the sys tem of wave equations

O:H i ~Hj a~ .--W -- ~+'.o'~ - - = O. ( 1 )

" Oxn Oxo

H

I

' a 6 e , . . . . . s a /0 .90 :tO ,~0 50 5~ zO aO 90/Ot"t, lO y :~0 x'~O t, lO sy

N ,, I

Fig. 1 Fig. 2

Fig~ 1. Ear th ' s field s trength as a function of t ime with discontinuities in

the field.

Fig. 2. Linear var ia t ion in Ear th ' s field during inversion with discontin- uities in the f i rs t derivative.

Kirov Polyteetmical Institute, Tomsk. Translated f rom Izvest iya Vysshikh Uchebnykh gavedenii, Fizika, No. 10, pp. 145-147, October, 1972. Original ar t ic le submitted June 30, 1971.

�9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 glest 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

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H

Fig. 3. Complex var ia t ion in Ear th ' s field during inversion, with possible discontinuities in h ighe r -o rde r der ivat ives and a per iod of zero field near the central inversion point.

This sys tem is a direct consequence of Maxwell 's equations for the e lect romagnet ic field in a nonideal conductor (taking into ac- count the conduction current and the displacement current) . The sub- scr ip ts and coefficients in (1) have the following signif icance: j = 1, 2, 3; K = 0, 1, 2, 3; x 1 =x , x 2 = y , x 3 = z, x 0 = t , H l = H x , H 2 = H y , H 3 =Hz , ajl = a j2= aj3 = 1 , a j 0 = - - ~ 0 e e 0 ; ~ and e are the relat ive mag- netic and e lect r ical p a r a m e t e r s of the mater ia l of the Earth, while ~0 and e 0 a re the same for vacuum, and 7 is the specific e lect r ical conductivity of the Earth (all quantities a re in SI).

We consider the Earth as a f i r s t approximation as a semicon- ducting homogeneous i sot ropic medium; the coefficients in (1) a re therefore constant. The charac te r i s t i c hypersur face S for the sys tem of the type of (1) is defined by

(xo, x~, x2, x~) = O, ( 2 )

and is a hypersu r face with discontinuities in the second derivat ives of the Hj [6]. It is assumed that the Hj and the f i rs t der ivat ives a re continuous. If there is a period of res t , we get the following approximate r e - sult

Mj 1 ~ o ~ H j - 2 [ 0_._~_~ "1~ ' (~ = o, 1, ), 3), (3) 0.~ \ ox,, ] ( j : 1, 2, 3),

which is obeyed near the charac te r i s t i c hypersur face (a s imi la r equation is readi ly derived from the equa- tions given on pages 497 and 498 of [6]).

H 1 Equation (3) shows that Aj~ = ~ is not dependent on j.

Then the f i rs t of these two concepts leads to AjK being independent of subscr ipt j. The second as - sumption enables us to find AjK from the formula for the field s t rength of a dipole; he re it is found that AjK is dependent on j in the dipole case, so at least one of these concepts must be incor rec t .

In principle, it is possible to abandon the assumption of a dipole charac te r for the source of the Ear th ' s magnetic field and to construct more complex hypotheses; but such hypotheses would be extremely art if icial , because they would require considerat ion of a very complex distribution pat tern for the current lines in the central par t s of the Earth. If we maintain the assumption about the dipole charac te r of the source (circular cur rents paral lel to the equator), we have to abandon the idea of a period of quiescence, and a s sume that the inversion curve runs very smoothly, so that discontinuities occur in the derivat ives of H only for values above the third. Of course, derivat ives of the third order and above can have such discontinuities, but continuity in the f i rs t and second derivat ives in turn conflicts with the usual conception that the inversion is sharp.

When one says that an inversion is sharp, one usually has in mind that the duration is much less than the interval of t ime between inversions; the duration of the p rocess in t ime units specifies the sharp- ness , and this in turn governs the order of the discontinuous der ivat ives . Then (3) enables one to estab- lish c r i t e r i a for the sharpness of the p rocess if one understands by the lat ter the order of the discontin- uous der ivat ives . The relat ive duration of the invers ion is being studied at present on experimental evi- dence; the question of the sharpness has so far not yet received appropria te investigation. Of course , it is possible to abandon the idea that the position of the dipole axis is constant, and thereby lose the need for a period of quiescence; but in that case in examining the deviation of the dipole axis f rom the axis of rotat ion we inevitably have to introduce sources of magnetic perturbat ions external to the mantle regions of the Earth and the radiation belts.

The considerat ion of such sources is [1, 3] devoid of physical in teres t , because it needs incorpora - tion of unknown cosmic factors ; however, this difficulty is hardly very ser ious , since the sources of the external fields may be not only cosmic fac tors , which have now been studied much better than in Frenkel ' s t ime, but also geological p rocesses in the Ear th ' s crust , such as earthquake fields, volcanic eruptions~ and so on.

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It would s e e m that a dipole sou rce for the magnet ic field of the Ear th is a mobi le sys t em, which to a f i r s t approximat ion is quas i l inear with two s tab le s t a tes . If the natura l f requency of the s y s t e m is com- pa rab l e to the mean f requency of the weak external per tu rba t ions (for ins tance, the magnet ic fields of ear thquakes) , the energy of the l a t t e r can accumula te in the sy s t em, increas ing the sca le of the smal l m o v e m e n t s about the equi l ibr ium posit ion, and causing thereby cons iderable deviations of the dipole axis or even comple te r e v e r s a l .

In fu r ther work on the theory of invers ion , one should give se r ious attention to the effects of smal l f luctuations in the externa l fields on the motion of the sou rce of the Ea r th ' s magnet ic field.

1, 2. 3. 4.

5e 6.

L I T E R A T U R E C I T E D

A. Cox, T. De l rymple , and R. Dowle, Usp. Fiz. Nauk, 94, No. 4, 719 (1968). Phys ics of the Ea r t h ' s Crus t and Upper Mantle [Russian t rans la t ion] , Mir , Moscow (1966). B. M. Yanovskii , T e r r e s t r i a l Magnet ism [in Russian] , GITTL, Moscow (1953). T. Rikimaki , E l ec t romagne t i sm and the In ternal St ructure of the Ear th [Russian t ranslat ion] , Nedra, Leningrad (1968}. Ya. I. F renke l ' , " T e r r e s t r i a l magne t i sm , " Izv. AN SSSR, Ser . Fiz., 11, No, 6, 607 (1947}. V. I. Smirnov, Textbook of Higher Mathemat ics [in Russian], Vol. 4, GITTL, Moscow (1957).

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