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First ZDB push. Testing repo functionality.
2024-08-12 13:51:37 -05:00
Downloaded from https://royalsocietypublishing.org/ on 05 February 2024
PHILOSOPHICAL TRANSACTIONS
OF THE
ROYAL SOCIETY OF LONDON
Series A. Mathematical and Physical Sciences No. 859 Vol. 243 pp. 67-92 27 October 1950
THE WESTWARD DRIFT OF THE EARTH’S MAGNETIC FIELD
By
E. C. Bullard, F.R.S., Cynthia Freedman, H. Gellman and J o Nixon
Published for the Royal Society by the Cambridge University Press
London: Bentley House, N.W. 1 New York: 51 Madison Avenue
Price Six Shillings
[ 67 ]
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T H E W E ST W A R D D R IF T O F T H E E A R T H ’S M A G N E T IC F IE L D
By E. C. BULLARD, F.R.S., CYNTHIA FREEDMAN, H. GELLMAN and JO NIXON
University of
Tornt, Scripps Institution of
,Ocea and National Physic
{Received 27 April 1950)
T he westward drift of the non-dipole p art of the earth’s magnetic field and of its secular variation
is investigated for the period 1907-45 and the uncertainty of the results discussed. It is found that
a real drift exists having an angular velocity which is independent of latitude. For the non-dipole
field the rate of drift is 0T8
±0-015°/year, that for the secular variation is 0-32 ±0-067°/year. T h
results are confirmed by a study of harmonic analyses made between 1829 and 1945.
T he drift is explained as a consequence of the dynam o theory of the origin of the earth’s field.
This theory required the outer part of the core to rotate less rapidly than the inner part. As a result
of electromagnetic forces the solid m antle of the earth is coupled to the core as a whole, and the
outer part of the core therefore travels westward relative to the mantle, carrying the minor features
of the field with it.
1. I ntroduction
It has long been known that many features of the earth’s magnetic field show a westward drift (Halley 1692). Recently Elsasser (1949) has emphasized the great theoretical interest of this phenomenon, which deserves a more careful examination than it has yet received. Such an examination meets with two main difficulties: first, the results obtained depend on what features of the field are examined; and secondly, it is difficult to estimate how far the apparent differences between the fields at two epochs are real, and how far they are due to the incompleteness of the data and to the diverse methods employed in reducing them to maps.
The latter difficulty is greatly reduced by the work of Vestine, Laporte, Cooper, Lange & Hendrix (1947 a). They have used virtually all the observations between 1905 and 1944 to give charts of the secular variation for 1912-5, 1922-5, 1932-5 and 1942-5. These charts have been used by them to reduce all the observations to 1945, and to construct tables and maps of all the components of the field for that year based on all the data. From this material we have computed the field for 1907-5, and our investigation is mainly based on the comparison of this and the field for 1945. In this procedure the fields compared are based on the same data and the spurious differences introduced in the reduction are as small as possible.
2. C omputation of th e non-dipole field
The earth’s field is roughly that of a dipole with its axis not far from the axis of rotation.
This predominant dipole field obscures the minor features, and it is desirable to remove its
effects before looking for the westward drift. Vestine et al. (1947 have determined the
dipole for 1945 and find its moment to have components i23 = —0-3057a3 parallel to the
earth’s axis, g\a? = —0-021 la3 at right angles to the axis and in the plane of the Greenwich
meridian and
h\a?— 0-0581a3 at right angles to these two directions in longitude 90° E,
Vol. 243. A. 859, (Price 6s.)
io
[Published 27 October 1950
68
E. C. BULLARD AND OTHERS ON THE
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where
ais the mean radius of the earth. The field, X, Y, Z remaining after subtracting that
due to the dipole
X ,d
Y,dZ d is called the non-dipole field and is g
X = X l —Xdi
Y = Y l —YdZ, = Z y~ Z
Xd = —sg°i in 0+g\ cos
-f h\ cos 6 sin <f>,
Yd — £} sin
6—ch\ os#,
Z d== —
g\co$0— sin 6c
Here X v Yx and Z x are the northerly, easterly and vertical (downward) components of the
field, 6 is the co-latitude and 0 the east longitude. X l} Yx and Z x are given by Vestine et
(1947
a,tables 49 to 51) and from them we have computed Y and Z. The results are given
in tables 1, 3 and 5. Figure 1 shows contours of the vertical component of the non-dipole
field and arrows representing its horizontal component. Figures 3 and 5 give contours of
the northerly and easterly components.' The contours have been drawn through points
obtained by plotting the data from tables 1, 3 and 5 for meridians and parallels spaced at
10° intervals. This yields the intersections of the contours with the lines of a 10° grid and
leave little arbitrariness in their form. In figures 1 and 2 the arrows representing the hori­
zontal force always point towards areas where the vertical force is a minimum and away
from those where it is a maximum. This behaviour is characteristic of a field whose origin lies within the earth.
The field for 1907-5 was obtained by subtracting the increase between 1907-5 and 1945
from the 1945 field. Vestine
et al.(1947 a, tables 24-35) give the rates of c
intervals between 1912-5 and 1942-5, call these xXi x2, x3, for the component and similarly
for the Y and Z components. The field AT{, Y{, Z[ in 1907-5 is then given approximately by
x [~ ,x ^ x v,
zi^ z.-z^
where
X v = 10*1+ 10*2+ 10*3+ 7-5*4,
Yv = 10^4-10^ + 10^ + 7-5^,
Z v= 1OZjl+ 10z2+ 1 0z3+ 7-5z4.
The approximation consists in putting the change over a 10-year interval equal to 10 times the rate of change at the central year of the interval. The error depends on the third and higher differentials of the field and is of no practical importance. A similar approximation has been made by Vestine in the construction of his charts. The non-dipole field for 1907-5, X ', Y', Z ' is most easily obtained by computing the dipole part X dv, Ydv, Zdvof the total secular variation and putting
X ' = X ; ~ X d+ X dv, Y ' = Y l - Y d+ Y dv,
The results are given in tables 2, 4 and 6 and figures 2, 4 and 6. Vestine et al (1947 b, tables 41 and 101 and p. 4) give a harmonic analysis of the secular variation from which we find the changes in the components of the dipole to be A = 0-0083, Ag} = 0-0008, Xh\ = -0-0018. By an oversight we used in computing X dv, Ydv and Zdv, not these values, but those obtained from an analysis of the same data using equal weights for all zones of latitude. These values were Ag® = 0-0086, A^} = 0-0022, Xh\ = 0-0004. Vestine’s values are to be preferred, but it was not thought worth while to repeat the work as the changes lie within the uncertainty of the data, and vary so slowly with position as to have no perceptible effect on the westward drift.
WESTWARD D RIFT OF THE EARTH’S MAGNETIC FIELD
69
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3. T h e w estw ard d rift of t h e non-d ipo le field
An examination of figures 1 to 6 clearly shows the westward drift. It is particularly well
seen in the positions of the maxima of vertical force over the Gulf of Guinea and in southern
Mongolia and in the zero line of east force off the west coast of Europe and of vertical, force
off the west coast of South America. On the other hand, the maximum of vertical force in
the eastern United States does not appear to have moved, and there are other features that
have changed so much in shape between the two epochs that it is difficult to decide whether
they have moved or not.
The determination of the westward drift by picking a few conspicuous features and deter­
mining their shifts from the maps is clearly an unsatisfactory procedure. It can be slightly
improved by the use of finite difference methods to find the maxima and other features.
A few results obtained by this method are given in table 7. These give a rough estimate of
the rate of drift (0-266°/year) and show the north-south component to be small. Since they
are based on the results for selected parts of the earth, little reliance can be placed on them,
and no estimate can be made of the uncertainty.
A more satisfactory method is to consider the values of the field at 10° intervals along
a circle of latitude and to determine what shift in the longitude of the 1907-5 field will make
it best fit the 1945 field. Let X{<j>) be any component of the non-dipole field in latitude 6 and
longitude
(j)in 1945 and X'{$) that in 1907-5. We now form
e = X(<f>)-X'(<!)+D)
for every 10° of longitude and for D = 30°, ±20°, ±10° and 0°. The e’s are the differences
between the field in 1945 and that in 1907-5 shifted D degrees to the west. The most probable
value of the shift is now taken as that which makes He2 a minimum, the summation being
over the 36 values of
e2on a single parallel of latitude. It may be shown that this procedur
is equivalent to choosing the shift that makes the correlation coefficient between the fields
in 1907-5 and 1945 a maximum. Separate estimates have been made for each component
of the field for every 20° of latitude, giving 27 determinations in all.
The work was entirely based on tables 1 to 6, and was done without reference to the maps.
The final results therefore follow uniquely from Vestine’s tables and, given those tables, are
independent of any smoothing or graphical procedures. The process may be illustrated by
the results for the vertical field on the equator. Figure 7 shows the fields for 1945 and 1907-5
as a function of longitude. The minimum of He2is found by computing (d/dD) He2at points
half-way between the points of tabulation, and finding the value of D for which it is zero by
inverse interpolation. In these and other finite difference processes fourth differences have
been retained where necessary. Figure 8 shows typical curves of He2as a function of the shift,
the points at ± 5 , ±15 and 25° were obtained by interpolation.
The twenty-seven estimates of the shift are given in table 8 together with their standard
errors derived in §4 below. Since all twenty-seven are to the westward, there can be no
doubt of the reality of the drift, which is established without recourse to the detailed
arguments of §4.
10—2
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>80° >90° 200° 210* 220* 230* 340* 2S0° 260° 270? 280° 290* 300° 310° 320° 330° 840° 350° 0 a
80* 90° 100° 110° 120° f30a \AO° 150° 160° 170° t80*
o
W O
. BULLARD AND OTHERS ON THE
180° 190° 20Q° 210° 220° 230° 240° 250° 260° 270* 280® 290° 300° 3T05 320° 330° - 340® 350° ~ 0 °
50° 60° 70° 80° 9 0' 100° JI0° 120° 130° I40‘ 150° 160° 170° 180°
Figure 1. Non-dipole field for 1945. T he contours give the vertical field at intervals of 0*02 gauss. T he arrows give the horizontal com ponent, an arrow 9*3 m m . long represents 0*1 gauss.
WESTWARD D RIFT OF THE EARTH’S MAGNETIC FIELD
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180* 190* 200* 210® 220* 230* 240* 250* 260' 270' 280® 290® 300' 310* 320* 330' 34 0 ' 350* 0* 10* 20* 30* 40* 50* 60* 70* 80* 90* 100* HO* >20* 130* 140* 150* 160* >70* >60*
Figure 2. Non-dipole field for 1907-5. T he contours give the vertical field at intervals o f 0-02 gauss. T he arrows give the horizontal com ponent, an arrow 9-3 mm. long represents 0-1 gauss.
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180® 190® 200® 210* 220® 230* 240® 250° 260* 270* 280* 290* 300° 310° 320® 330® 340® 350®
20* 30®
140® 150® 160® 170*
<1
to
W o
. BULLARD AND OTHERS ON THE
Figure 3. Non-dipole field for 1945, north com ponent, contour interval 0-02 gauss.
Downloaded from https://royalsocietypublishing.org/ on 05 February 2024
180° 190° 200a 210° 220* 230° 240° 250° 260*
310* 320° 330° 340° 350°
M0° >20* >30°
WESTWARD D RIFT OF THE EARTH’S MAGNETIC FIELD
180* 190* ZOO4
Figure 4. Non-dipole field for 1907*5, north component, contour interval 0*02 gauss.
co
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160* >90* 200* 210* 220* 230* 240* 250* 2604
320* 330* 340* 350*
120* 130* 140* 150* 160* 170* 180*
<1
. BULLARD AND OTHERS ON THE
180° 190* 200* 210* 220* 230* 240* 250* 260* 27CT 280* 290s 300* 310* 320* 330* 340‘
Figure 5. Non-dipole field for 1945, east com ponent, contour interval 0*02 gauss.
120* 130* 140* 150* 160* 170* 180*
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180* 190* 200* 210* 220* 230* 240* 250* 260* 270* 280° 290* 300* 310*
WESTWARD D R IFT OF THE EARTH’S MAGNETIC FIELD
° 2L0° 220* 200
230° 240* 250® 260* 270° 280* 290* 300* 310* 320* 330‘
Figure 6. Non-dipole field for 1907-5, east component, contour interval 0-02 gauss.
01
76
E. C. BULLARD AND OTHERS ON THE
lat.
80° N E long.\
0° 10° 20° 30° 40°
50° 60° 70° 80° 90° 100° 110° 120° 130° 140° 150° 160° 170° 180° 190° 200° 210° 220° 230° 240° 250° 260° 270° 280° 290° 300° 310° 320° 330° 340° 350°
37 28 17
3 -1 0 -2 6 -4 2
-6 1 -7 4 -8 3 -8 6 -8 8 -8 5 -7 9 -7 1 -6 4 -5 5 -4 8 -4 1 -3 5 -2 9 -2 6 -2 0 -1 8 -1 6 -1 2 -1 0 - 4
3 8 18 26 33 37 41 43
T able 1. N on-d ipo le fie l d , n o r th com ponent, 1945, 10~3 gauss
70° 60° 50° 40° 30° 20° 10° 0°
10° 20°
30°
40° 50°
22 15 5 - 9 -2 3 -4 0 -5 5
-6 9 -8 3 -9 0
-9 3 -9 3 -8 9 -7 9 -6 4 -4 7 -3 4 -2 2 -1 2 -1 0 - 9 -1 3 -2 3 -3 4 -4 0 -4 3 -4 1 -3 9 -3 3 -2 3 - 6 - 2
8 17 24 24
6 7 - 5 -1 2 -2 5 -3 6 -4 6
-5 7 -6 6
-7 3 -7 6 -7 4 -61 . -4 6 -3 4 -2 2 - 5
5 13 10
4 0 -1 0 -1 8 -2 8 -3 8 -4 4 -4 8 -4 7 -4 1 -2 7 -1 4 - 4 3 12 12
10 7 1
- 8 -1 3 -2 0 -31 -3 4
-3 7 -3 7 -3 4 -3 0 -2 7 -1 8
- 9 0 5 11 12
9 6 3 4
- 3 -1 0 -1 7 -2 7 -3 6 -3 7 -3 5 -2 8 -1 7 - 6
2 6 8
17 19 15 12 5
1 0 0 6 14 15 14 15 13 12 11 8 3 - 1 - 1
0 4 7 10 11 6 - 1 -1 1 -1 9 -2 4 -2 4 -1 7 - 8 5 12 15
34 36 32 34
33 33 34 41 46 56
59 55 47 37 26 19
8 0 - 6 - 8 - 6 0 9 19 22 26 21 10 - 1 - 9 -1 8 -2 0 -1 2 2 14 27
36 42
45 49 52 56 61 69 80 87
89 82 69 55 39 28 13
3 - 3 - 5 - 4
3 13 22 28 35 34 28 15 3 -1 1 -1 5 -1 4 - 3 12 23
15 25 32 40 46 55 64 73 85 95 99 91 81
70 55 40 31 23 15 14 13 16 21 24 29 33 31 26 17
8 - 7 -1 5 -1 6 -1 2 - 3
5
-3 0 -2 1 -1 2
3 12 21 36 50 63 79 83 85 82 75 66 58 51 43 40 36 30 25 24 22 18 22 20 9 8 - 1 -11 -21 -3 1 -3 6 -3 6 -3 3
-7 9 -7 3 -6 7 -5 5 -4 2
-2 9 -1 3
7 22 42 54 67 68 72 71 69 62 58 53 47 38 27 18 12 4
1 - 3 - 5 -1 3 -21 -3 2 -4 3 -5 7 -6 5 -7 1 -8 4
-117 -113 -112 -105 - 95 - 79 - 55 - 41 - 22 - 1
21 38 51 57 64 58 55 5Q 47 38 30 21 13 2 - 4 - 15 - 21 - 29 - 34 - 42 - 52 - 64 - 76 - 93 -103 -115
-127 -130 -1 2 9 -1 2 6 -113 - 95 - 82 - 67 - 54 - 35 - 15
5 21 30 34 35 34 32 30 25 18 14 9
1 - 10 - 16 - 27 - 37 - 43 - 51 - 59 - 73 - 86 -102 -110 -119
-1 0 8 -111
-110 -103 - 96 - 82 - 71 - 62 - 57 - 49 - 38 - 27 - 10 - 1
6 9 11 14 13 11 7 7 5 0 - 7 - 13 - 23 - 30 - 34 - 38 - 44 - 55 - 69 - 80 - 92 -101
-6 0 -6 6 -6 5 -5 6 -4 9 -41 -3 5 -3 6 -4 0 -3 8 -3 8 -3 8 -3 2 -2 5 -1 9 -1 8 -1 2 -11 - 3 - 2 - 1
0 2 0 - 4 - 7 -1 0 - 7 - 7 - 5 - 8 -1 4 -2 3 -3 4 -4 6 -5 3
60°
- 9 -1 5 -1 6 -1 2 - 5
6 7 2 - 7 -1 8 -2 3 -3 2 -3 9 -4 3 -4 5 -4 5 -4 0 -3 5 -3 1 -2 6 -2 0 -1 8 -1 5 -1 4 -1 0 - 5 0 9 17 27 32 33 26 16 6 - 2
70° 80° S
38 34 35 41
44 44 37 26 12 -1 1 -3 0 -3 7 -3 8 -4 1 -5 1 -6 9 -7 8 -6 6 -6 0 -5 5 -4 9 -4 6 -4 3 -3 8 -31 -2 3 -1 2
1 12 24 45 60 67 67 58 47
90 82 73 61 49 33 17 - 2 - 27 - 44 - 58 - 72 - 82 - 88 - 94 -101 -102 - 99 - 93 - 93 - 85 — 78 - 70 - 64 - 55 - 44 . - 27 - 6 11 28 44 60 73 85 93 93
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T able 2. N on-d ipo le field , no rth component, 1907-5, 10“3 gauss
\
80° N 70° 60° 50° 40° 30° 20° 10°
0°
10°
20°
E long.X^
30° 40° 50° 60° 70° 80° S
0° 10°
20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 140° 150° 160° 170° 180° 190° 200° 210° 220° 230° 240° 250° 260° 270° 280° 290° 300° 310° 320° 330° 340° 350°
41
24
3
2
1
13
18
4 -3 4 -7 6 -105 -108 -8 2 -41
0
37
78
34
20
9
3
7
17
24
13 - 2 6 - 7 0 - 1 0 2 - 1 1 0 - 8 8 - 5 2 - 1 1
29
71
25
13
2
2
8
16
28
20 - 1 7 - 6 4 - 99 - 1 0 7 - 9 2 - 5 7 - 1 8
26
63
13
3 -1 -2
9
22
34
28 - 2 - 5 1 - 91 - 1 0 6 - 9 1 - 5 4 - 2 0
28
53
1 -10 -10 - 3
12
26
38
34
7 - 3 9 - 83 - 94 - 8 9 - 5 2 - 1 9
26
43
-14 -26 -20 - 8
4
27
43
41
13 - 2 8 - 70 - 85 - 7 7 - 4 6 - 1 1
26
30
-29 -41 -31 -18
4
28
47
47
24 - 1 7 - 49 - 71 - 6 4 - 3 8 - 1 0
20
18
-48 -56 -43 -21
3
33
52
52
34 - 1 - 38 - 54 - 5 1 - 3 6 - 1 2
12
4
-62 -70 -53 -27
5
35
60
60
43
11 - 21 - 40 - 4 3 - 3 6 - 1 7
3 -17
-71 -78 -62 -31
10
42
65
71
58
29 - 3 - 22 - 3 4 - 3 3 - 2 5 - 1 5 - 3 1
-76 -83 -67 -32
9
44
69
78
65
41
17 - 6 - 2 5 - 3 2 - 2 8 - 3 1 - 4 2
-79 -84 -68 -30
6
42
65
74
71
55
33
8 -18 -34 -36 -35 -53
-78 -81 -57 -28
8
35
55
67
72
60
47
20 - 6 - 3 1 - 4 0 - 3 2 - 6 0
-73 -73 -43 -20
6
26
42
58
68
67
54
28 - 1 - 2 6 - 4 1 - 3 3 - 6 6
-67 -59 -32 -11
6
16
27
45
60
68
62
33
4 -19 -40 -41 -72
-62 -44 -21 - 3
5
9
16
30
52
67
57
36
9 -16 -38 -57 -82
-54 -33 - 5
1
1 -1
2
22
45
60
56
36
12 - 8 - 3 2 - 6 5 - 8 6
-49 -22
3
6 -3 -9 -7
15
38
58
53
36
17 - 6 - 2 5 - 5 3 - 8 6
-44 -14
9
7 - 7 -13 - 9
9
37
54
52
36
17
3 -22 -49 -84
-40 -13
5
4 - 6 -13 - 9
11
35
50
45
33
17
4 -19 -47 -88
-36 -14 - 2
1 -3 -8 -6
12
31
43
38
27
15
4 -16 -44 -84
-34 -20 - 6 - 1
2
0
3
17
28
33
31
24
15
4 -17 -44 -81
-29 -30 -17
0
6
10
15
23
28
25
23
20
13
5 -16 -45 -76
-28 -42 -26 - 7
10
22
25
27
26
19
13
12
8
3 -16 -43 -74
-27 -49 -37 -14
12
27
33
33
21
10
7
1
2 - 1 -11 -39 -69
-24 -53 -47 -21
9
34
43
38
24
5 - 6 - 5 - 3 - 1 - 5 -32 -62
-22 -52 -54 -30
4
34
45
38
21 - 2 - 16 - 17 - 1 1 - 2
3 -21 -48
-17 -51 -59 -39 - 1
26
43
35
9 - 8 - 27 - 29 - 1 6
3
14 - 6 - 2 9
-10 -46 -59 -40 - 8
18
32
25
4 - 2 0 - 35 - 36 - 1 9
6
25
7 -13
- 4 -37 -55 -40 -15
8
21
15 - 7 - 3 0 - 44 - 44 - 2 1
13
38
21
5
7 -20 -42 -37 -21 - 7
2 - 2 - 1 6 - 4 0 - 52 - 50 - 2 4
14
46
44
22
18 - 1 5 - 3 0 - 2 9 - 2 1 - 1 8 - 1 0 - 1 4 - 2 5 - 4 6 - 60 - 60 - 3 1
14
50
62
39
27 - 2 - 1 8 - 2 0 - 1 9 - 1 9 - 1 8 - 2 1 - 3 3 - 5 5 - 68 - 70 - 4 3
7
46
70
54
33
9 - 9 - 1 3 - 1 1 - 1 3 - 1 4 - 1 9 - 3 7 - 6 0 - 83 - 84 - 5 3 - 4
36
70
68
40
19
2 - 8 - 7 - 7 - 5 - 1 2 - 3 8 - 6 5 - 91 - 91 - 6 5 - 1 8
23
61
77
41
22
6 -3 -2
4
5 - 6 -3 6 -7 9 -102 -100 -7 4 -2 9
11
49
79
WESTWARD DRIFT OF THE EARTH’S MAGNETIC FIELD 77
\ lat.
\ o n g .\
0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 140° 150° 160° 170° 180° 190° 200° 210° 220° 230° 240° 250° 260° 270° 280° 290° 300° 310° 320° 330° 340° 350°
80° N 70°
42 52 62 70 75 78 76 76
64 48 31 13
0 — 12 -2 3 -3 3 -3 9 -4 2 -4 2 -4 2 -41 -4 0 -3 9 -3 9 -3 7 -3 5 -3 7 -3 7 -37 -3 4 -33 -27 -15 - 3
10 31
33 49 62 73 82 87 84 78 65 48 27
2 -2 3 -4 2 -5 0 -5 7 -5 5 -5 0 -4 0 -2 9 -2 0 -1 3 - 7
-1 0 -1 3 -21 -2 9 -3 5 -4 3 -4 6 -5 4 -4 0 -32
-23 - 4
16
T able 3. N on-dipole field , east component, 1945, 10-3 gauss
60° 50° 40° 30° 20° 10° 0° 10° 20° 30° 40° 50°
30 45 62 75 82 89 84 78 65 45 20 - 8 -3 7 -61
-6 6
-7 1 -6 4 -51 -3 7 -1 8 - 4
7 11 15 12 - 3 -16 -3 4 -4 7 -5 2 -5 4 -5 3 -43 -2 7 -11 11
27 45 61 71 78 84 79 72 59 37 11 -1 6 -4 5 -63 -7 4 -7 0 -61 -4 6 -2 9 -11
0 8 24 26 24 12 - 3 -2 0 -3 9 -51 -5 8 -5 7 -5 0 -3 4 -1 3 8
22 42 58 69 76 76 71 60 45 27
5 -2 0 -4 0 -5 7 -6 7 -6 5 -5 2 -3 7 -1 8 - 7
4 13 22 25 26 20 11 - 8 -29 -4 4 -5 5 -5 9 -5 4 -36 -1 9 3
17 37 54 64 69 69 60 48 32 17 4 —14 -2 9 -4 5 -5 3 T 51 — 38 — 24 -1 2 - 7 - 1 5 11 16 22 22 17
7 -11 -31 -5 0 -5 8 -57 -4 6 -2 8 - 6
10 31 48 58 64 62 50 36 19
9 4 - 3 -1 4 -2 7 -3 5 -3 4 -2 5 -1 4 - 8 - 5 - 8 - 2 3 8 15 18 20 17 2 -1 6 -41 -5 4 -6 2 -5 9 -41 -1 4
- 1 23 41 52 56 55 39 21 10 5 9 6 1
-1 0 -1 9 -1 9 -1 0 - 8 - 4 - 6 -1 0 -1 2 - 4
2 9 18 25 26 17 - 4
-3 1 -4 9 -63 -67 -5 4 -2 9
-1 2 13 30 41 48 47 25 6
- 5 - 1
11 11 7 0 - 6 - 8 - 4 3 0 - 7 -1 0 - 7 - 3 - 5 6 18 29 31 24 6 -21 -4 4 -5 8 —66 -6 0 -3 9
-21 0 19
31 39 32
9 -1 0 -19 -1 2
3 10 7 3 - 2 - 4 1 3 5 1 - 1 0
1 4 11 19 30 39 32 14
- 9 -3 5 -5 0 -5 6 -5 5 -3 9
-2 4 - 6
10 . 23
25 11 -1 0 -31 -4 0 -3 6 -15 - 3 1 0 0 2 5 10 11 10 7 9 11 15 18 27 36 44 40 28 2 -2 2 -3 9 -4 4 -4 4 -35
-21 - 7
6 17 13 - 4 -29 -5 0 -6 2 -56 -41 -2 6 -11 - 5 - 1 4 12 14 14 17 17 19 22 26 30 39 49 54 54 38 15 - 9 -2 3 -3 0 -3 3 -2 9
-17 - 9
4 6 - 1 -1 9 -4 7 -6 8 -7 8 -8 0 -6 6 -5 2 -3 2 -1 6 - 6 1 9 16 19 22 24 28 30 35 44 52 65 70 65 53
30 8
- 9 -1 8 -2 2 -2 4
-1 6 - 9 - 4 - .7 -1 9 -37 -6 5 -8 6 -9 6 -9 4 -87 -71 -51 -2 9 -19 - 6
5 17 20 28 32 37 43 52 59 68 82 86 79 69 46 24 7 - 5 -1 3 -16
60°
- 12 - 15 - 17 - 31 - 53 - 72 - 91 -103 -111 -102 -101 - 92 - 57 - 39 - 28 - 13
1 16 20 28 33 39 51 62 77 88 97 98 92 81 64 43 27 15 0 - 8
70°
- 24 - 39 - 55 - 75 - 96 -110 -118 -122 -127 -116 -104 - 82 - 55 - 49 - 46 - 35 - 23 - 2
13 19 28 40 56 - 72 91 108 119 114 101 95 89 65 47 30 13 - 10
80° S
- 39 - 64 - 86 -103 -115 -122 -127 -132 -130 -114 -101 - 96 - 88 - 73 - 56 - 37 - 19 - 2
15 29 41 53 66 88 104 113 121 117 108 98 86 73 59 37 15 - 12
Downloaded from https://royalsocietypublishing.org/ on 05 February 2024
\ lat.
\ E long.\
0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 140° 150° 160° 170° 180° 190° 200° 210° 220° 230° 240° 250° 260° 270° 280° 290° 300° 310° 320° 330° 340° 350°
80° N
30 41 53 63 71 76 77 79 69 55 40 23 11
0 -1 0 -2 0 -27 -31 -3 2 -33 -3 4 -3 4 -3 5 -3 6 -36 -3 6 -4 0 -4 1 -43 -4 2 -43 -39 -2 8 -1 7 - 4
18
T able 4. N on-dipole field , east component, 1907-5, 10-3 gauss
70° 60° 50° 40° 30° 20° 10° 0° 10° 20° 30° 40° 50° 60° 70° 80° S
18
11
5
0 - 4 - 1 0 - 2 0 - 2 8 - 3 6 - 3 9 - 3 9 - 3 6 - 2 8 - 1 7 - 16 - 15
35
27
22
18
13
7
1 - 7 - 1 9 - 2 6 - 3 0 - 3 1 - 2 4 - 2 1 - 30 - 40
51
47
42
34
28
23
16
7 - 3 - 1 3 - 2 0 - 1 9 - 1 9 - 2 2 - 46 - 61
66
64
57
49
43
36
29
18
9
0 - 7 - 1 3 - 1 8 - 3 4 - 64 - 78
77
76
68
64
56
50
41
32
23
11
1 - 7 - 2 3 - 5 0 - 82 - 89
85
87
81
73
65
57
50
43
33
17
4 - 1 1 - 2 9 - 6 3 - 94 - 95
85
87
83
75
64
55
46
37
25
11 - 6 - 2 7 - 4 7 - 7 5 - 99 -1 0 1
82
84
81
70
59
48
36
25
13 - 3 - 2 2 - 4 1 - 6 2 - 8 1 -1 0 1 -1 0 7
71
75
72
61
48
37
28
13
1 -1 8 -3 8 -5 4 -7 3 -8 8 -106 -108
56
56
52
46
37
28
21
10 - 1 - 2 3 - 4 2 - 6 4 - 7 6 - 8 2 - 98 - 95
37
32
26
22
19
18
19
16
4 - 1 3 - 3 5 - 5 6 - 7 4 - 8 6 - 89 - 88
14
5 -2 -7 -3
7
11
12
7 - 8 - 2 8 - 4 9 - 6 3 - 8 0 - 72 - 90
-10 -24 -32 -30 -22 - 9
3
7
5 - 4 - 1 7 - 3 3 - 4 5 - 4 9 - 50 - 88
-29 -49 -52 -49 -39 -23 - 8
1
3 - 2 - 8 - 1 7 - 2 6 - 3 5 - 49 - 79
-37 -55 -64 -58 -45 -29 -15 - 3
1
2 - 1 - 8 - 1 9 - 2 7 - 50 - 67
-45 -60 -61 -58 -45 -29 -14 - 3
1
5
5
1 - 7 - 1 7 - 44 - 33
-44 -53 -53 -46 -33 -22 - 8 - 2
4
6
12
7
2 - 7 - 39 - 40
-40 -43 -40 -33 -22 -13 - 9
2
2
8
11
12
10
3 - 22 - 28
-31 -30 -25 -16 -12 - 9 - 7 - 4
1
7
8
11
7
2 - 11 - 15
-21 -13 - 8 - 6 - 9 - 8 -11 -13 - 7
2
7
10
11
7-8-4
-1 4
0
1
4 - 3 -12 -16 -18 -10 - 4
4
9
12
10
1
7
- 8
9
8
12
2 - 7 -19 -16 -10 - 3
5
12
18
17
14
21
- 4
12
24
21
7 - 3 -12 -12 - 9
0
9
16
26
31
31
36
- 8
17
27
25
13
2 - 6 -14 - 4
5
16
24
38
44
50
61
-1 1
15
27
27
20
9
1 -3
3
10
23
37
50
63
72
80
-19
1
17
23
20
12
9
8
11
21
35
49
63
79
93
93
-28 -11
2
14
15
13
15
18
22
32
49
67
82
94
109
105
-36 -30 -15 - 7
5
9
15
22
34
44
58
77
92 100
109
106
-45 -46 -36 -27 -11
0
14
23
34
46
63
77
90
98
101
102
-51 -5 4 -50 -39 -23 - 4
9
22
30
45
55
70
84
91
99
97
-62 -60 -60 -51 -36 -17 - 1
9
19
27
37
49
63
76
97
90
-51 -61 -63 -59 -48 -34 -21 -12 - 7
0
10
25
39
55
76
82
-45 -56 -60 -58 -53 -49 -4 2 -35 -31 -25 -13
0
17
37
59
72
- 3 8 - 4 3 - 4 7 - 4 4 - 4 8 - 5 4 - 5 6 - 5 4 - 4 7 - 3 8 - 2 8 - 1 7 - 3 20
41
53
-20 -29 -29 -32 -37 -46 -55 -63 -55 -46 -38 -28 -17
1
22
34
0 - 6 -11 -15 -22 -29 -41 -48 -48 -44 -41 -36 -25 -10 - 1
10
I 1-2
78
E. G. BULLARD AND OTHERS ON THE
lat.
E lo n g .\
0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 140° 150° 160° 170° 180° 190° 200° 210° 220° 230° 240° 250° 260° 270° 280° 290° 300° 310° 320° 330° 340° 350°
T able 5. N on-d ipo le fie l d , v ertica l (dow nw ards) com ponent, 1945, 10-3 gauss
80° N 70° 60° 50° 40° 30° 20°
10°
0°
10° 20° 30°
40°
50°
60° 70° 80° S
-9 0 -7 4 -7 0 -6 2 -5 4 -4 4 -3 5 -2 4 -1 5
- 9 - 6 - 4 - 3 - 3 - 5 - 8 -1 2 -1 7 -2 2 -2 5 -2 8 -3 0 -3 2 -3 8 -3 7 -4 2 -4 8 -5 3 -6 1 -6 8 -7 2 -7 9 -8 7 -9 3 -9 7 -9 6
-8 8 -7 9 -6 9 -5 6 -4 3 -2 2 - 1
15 34 49 65 79 78 58 38
30 15
1 - 9 -1 6 -2 0 -1 8 -1 4 -1 0 - 9 -1 5 -2 4 -3 3 -4 1 -4 8 -5 7 -6 9 -8 3 -9 4 -9 7 -9 3
-8 2 -5 9 -5 6 -3 8 -1 7
11
50 65 91 113 125 129 124
109 78 51 23
6 0 -1 1 -1 2 - 8 - 2 14 23 27 26 20 1 -1 4 -3 7 -5 5 -71 -8 4 -8 8 -91
-6 9 -5 6 -3 7 -1 8
6 41 77 102 127
150 162
163 152 117
84
50 16 -1 0 -2 4 -2 9 -2 6 -1 0
3 20 36 51 56 55 39 18 - 9 -2 5 -5 5 -6 7 -8 2 -7 9
-6 4 -4 9 -31 - 6
27 61 93 124 144 161 169 169 152
118 80 37
6 -1 6 -2 5 -2 9 -2 4 -1 6 - 4
17 33 50 58 66 61 39 16 -1 7 -4 7 -5 9 -6 8 -7 3
-7 1 -5 8 -3 8 - 8
25 56 89 112 127 137 142 137 128 101 70 38 10 - 7 -1 4 -1 6
-1 5 - 6
1
9 22 34 48 62 63 51 *30 - 4 -3 6 —66 -7 7 -8 2
-102 - 86 - 72 - 42 - 6
24 48 80 76 84 88 90 92 79 56 34 17 11
5 4 3 - 2 - 4 - 4 1 17 31 43 48 51 31 7 - 32 - 65 - 93 -107
-1 4 0 -1 2 9 -109 - 87 - 55 - 23 - 9
21 19 24 24 40 48 48 40 33 24 25 19 12 11
5 - 10 - 18 - 22 - 12
- 3 14 32 39 32 13
- 25 - 67 -104 -134
-155 -151 -139 -117 - 89 - 57 - 32 - 18 - 34 - 37 - 31 - 16
4 11 7 8 8 17 10 9
1 - 8 - 20 - 33 - 37 - 39 - 31 - 4
15 26 27
9 - 23 - 65 -106 -140
-124 -130 -128 -111 - 89 - 61 - 48 - 33 - 70 - 79 - 75 - 58 - 45 - 30 - 30 - 29 - 27 - 22 - 20 - 22 - 17 - 27 - 32 - 37 - 39 - 48 - 33 - 12
9 26 30 17 - 11 - 44 - 77 -106
-5 9 -7 0 -6 8 -5 7 -3 9 -3 0 -21
-2 9 -5 6 -8 6 -9 4
-9 3 -8 2 -7 6 -7 5 -6 9 -6 5 -6 5 -5 4 -4 9 -3 9 -3 8 -3 9 -3 5 -3 7 -3 7 -2 2 - 7
22 45 52 47 29
2 -21 -41
22
98
14
95
20 100
26 102
45 107
50 105
49 101
24 • 73
- 19
34
- 61 - 20
- 95 - 61
- 1 0 3 - 86
- 1 0 2 - 99
-107 -109
- 99 - 1 1 4
- 98 - 1 1 0
- 89 - 1 0 3
- 86 - 94
- 75 - 82
- 69 - 72
- 57 ' - 51
- 49 - 51
- 41 - 47
- 34 - 37
- 30 - 26
- 23 - 9
- 7
7
18
46
53
88
77 114
93
134
94 144
85 142
68 133
51
120
35 108
146 147
150 152 157 158 144
113 74 28 - 22 - 67 - 95 -106 -111 -113 - 99 - 96 - 87 - 81 - 75 - 59 - 48 - 34 - 26 - 6 30 61 98 130 154 167 171 168 162 157
171 171
169 160 147 137 121 98 74 35 - 13 - 37 - 62
- 89 - 96 -108 -109 -110 -103 - 90 - 81 - 78 - 64 - 53 - 40 - 16
23 59 91 115 138 154 165 171 173 172
133 127 120 109 96 84 70 55 38
16 - 6
-2 7 -4 7 -6 6 -81
-9 0 -91 -8 6 -7 8 -7 2 -6 7 -6 2 -5 1 -3 6 -1 6
9 33 58 79 95 109 117 123 130 134 136
75 71 66 61 54 46 36 26 16 5 - 8 -1 9 -3 0 -37 -42 -4 4 -4 6 -4 6 -4 7 -4 4
-4 3 -3 9 -2 6 -1 6
- 3 12 24 35 47 56 63 69 73 75 76 76
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T able 6. N on-d ipo le field , v ertical (dow nw ards) com ponent, 1907-5, 10“3 gauss
80° N 70°
60° 50° 40° 30° 20° 10°
0°
10° 20° 30° 40° 50° 60° 70°
- 98 - 83 - 79 - 72 - 63 - 54 - 45
- 34 - 25 - 19 - 16 - 13 - 11 - 11 - 12 - 14 - 16 - 20 - 24 - 26
- 29 - 31 - 33 - 38 - 36 - 41 - 47 - 52 - 61 - 68 - 73 - 81 - 90 - 97 —103 —103
-102 - 99 - 90 - 73 - 59 - 37 - 17 - 2
17 33 51 67 66 45 27 21
8 - 3 - 10 - 15 - 20 - 19 - 14
- 8
- 5
- 10
- 18 - 25
- 33 - 41 - 52 - 67 - 84 - 98 —103 —103
- 98 - 81
- 80 - 64 - 43
14 25 40 67 92 107 114 108 90 60 38 15
2 0 - 7 - 7
- 6 0 17
28 34 36 33 16
1 - 25 - 47 - 66 - 84 - 96
-101
-8 5 -7 7 -6 6 -5 4 -3 2
3 39 66 95 123 140 143 136 99 67 39 11 -11 -20 -22 -16 - 4
9 26 45 64 72 72
59 39
10 - 9 -4 3 -61 -82 -8 5
-7 2 -6 2 -5 6 -4 0 -1 5
12
43 78 105 130 145 152 138 104 70 32
5 -1 3 -1 8 -1 8
-1 0
- 8 2
23 41 63 73 82 77 57 36
5 -27 -46 -63 -7 7
-6 8
-6 7 -5 9 -4 2 -1 7
8
40 68 91 111 125 127 119
93 65 37 13
0
- 4
-- 31 1
7 14 28 42 54
66
66 55 41 15 - 9 -3 9 -57 -6 9
-8 7 -8 5 -8 2 -6 2 -3 8 -1 5
8 48 52 69 82
88
90 76
56 38 24 21 18 19 18 13
21
5 17 25 30 31 37 27 19 - 4
-3 0 -5 9 -8 0
-113 -117 -106 - 95 - 78
53 35 3 11 25 30 46 53 51 45 41 36 40 36 30 25
11 4 13
20 15 16 8 2 11 26 20
5 27 62 96
-1 2 1 - 88
-129 -103
-127 ~ -111
-115 -107
-104 -101
- 81
81
- 54
56
- 21
27
- 24
50
- 19
48
- 12
45
0
37
15
30
20
17
16
18
21
15
24 36
91
31
2
27
2
15
4
- 1
21
- 15
29
- 29
35
- 35
38
- 44
51
- 46
45
- 32
32
- 17
15
- 3
8
14
29
19
27
. 10
23
- 23
2
- 61
33
- 97
64
-2 8 -4 7 -5 8
-6 5 -6 0
t-54
-31 -2 4 -3 6 -5 4 -6 2
-6 9 -6 4 -6 2 -61 -5 5
-47 -4 5 -3 3 -3 1 -27 -3 4
-4 0 -3 8 -4 0 -4 0 -27 -17
8 36 63 70 60 42
19 - 4
42 23 12
3 12 15 24 17 -11
-4 0 -6 9 -7 9 -8 3 -9 1 -8 5 -8 4 -7 2
— 68 -5 8 -5 6 -5 0 -5 1 -4 8 -4 4 -4 0 -3 3 -1 5
11 40 66 104 113 110 101 83 63
103 137 152 112
88 129 149 108
77 122 143 102
69 117 131
92
66 117 117
81
63 118 109
72
65 109
98
62
53
89
83
53
29
62
68
40
- 12
28
32
25
■ 46 - 1 4 - 3
8
• 69 - 5 5 — 22 - 8
• 82 - 8 1 - 4 3 - 2 5
■ 94 - 9 0 - 6 7 - 4 1
100 - 9 4 - 7 2 - 5 6
■ 96 - 9 5 - 8 3 - 6 7
• 88 - 8 1 - 8 7 - 7 1
• 78 - 7 9 - 9 2 - 6 9
• 69 - 7 6 - 9 1 - 6 6
- 64 - 7 6 - 8 6 - 6 6
■ 50 - 7 7 - 8 4 - 6 7
- 58 - 7 0 - 8 9 - 7 0
• 61 - 6 7 - 8 3 - 6 6
■ 56 - 6 0 - 8 0 - 5 8
• 47 - 5 7 —74 - 4 4
• 28 - 3 8 - 5 5 - 2 4
■ 10 - 2 - 1 9 - 3
30
30
17
20
66
65
50
40
98 104 78 57
137 139 106
73
153 159 127
84
156 167 142
93
151 167 151 103
138 160 155 110
121 153 155 114
80° S
61 59 56 54 50 44 35 30 22 14 4 - 5 -1 3 -22 -2 7 -30 -3 3 -3 6 -39 -3 8 -4 2 -41 -3 4 -2 8 -1 8 - 6 4 13 24 33 40 47 52 55 58 60
WESTWARD DRIFT OF THE EARTH’S MAGNETIC FIELD
79
T able 7. W estward d rift of selected features
place
Gulf of Guinea Gulf of Guinea South Mongolia England Brazil Turkey
feature
min. of vertical zero of horizontal max. of vertical zero of east zero of east zero of vertical
position 1945
____________ A
lat.
long.
0°-9N 5-6N 43-2N 50 N 0 40 N
2°-0E 2-7E 105-8 E 13-8 W 67-6W 46-9E
shift 1907-5 to 1945
A
lat.
long.
0°-lN 0-5N 0-8N
—
—
—
10°-6W 8-5 W 3-8W 10-7 W 12-2 W 14-4W
m ean 10-0 W
Downloaded from https://royalsocietypublishing.org/ on 05 February 2024
non-
1907-5 1945
U1 0 02
-Sjooi
160° 2 0 0 ° 2 4 0 ° 280°
east longitude Figure 7. V ertical field on the equator.
360° 40°
3 0 ° 20° IO° O" IO° 2 0 °
Figure 8. V ariation of 2 e 2 with shift, east com ponent, northern hemisphere. The figures on the right m ark the zero lines for the curves referring to 0, 20, 40, 60 and 80° N latitude.
T a ble 8. W estw ard d r ift 1907-5 to 1945
lat.
80° N 60 40 20 0 20 40 60 80S
vertical
5°-2 + 5°-4 4-2+ 2-8 5-6+ 2-5 9-0+ 1-7 8-9+ 2-2 10-1+2-9 2-8+ 2-7 6-7+ 4-1 13-9 ±8-5
mean 7-3+ 0-9
east
8°-4 + 3°-5 4-7+ 1-6 5-9+ 1-5 9-1+ 1-7 9-9+ 1-7 10-5 + 1-9 7-0+ 2-7 3-8+ 3-4 10-7 + 6-0
7-3 ±0-7
north
8°-4 + 4°-l 4-2 + 3-5 11-9 + 4-7 4-0+ 3-7 4-1+ 2-5 1-6+ 1-7 6-9+ 5-1 1-5+ 5-5 8-9 ±5-9
4-7 ±1-1
mean
7°-8 + 2°-4 4-5+ 1-3 6-3+ 1-2 8-7+ 1-1 8-3+ 1-2 6-3+ 1-2 5-1 + 1-8 6-8+ 2-4 10-6 ±3-8
6-74 ±0-49
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80
E. C. BULLARD AND OTHERS ON THE
4. T he uncertainty in the determination of the drift
The experimental errors in the determination of the magnetic field are always negligible for our purpose and the errors of reduction to the epochs 1907*5 and 1945 are usually so. The field is, however, subject to local anomalies that frequently reach 0*02 gauss. In the pre­ paration of his charts Vestine has smoothed out all but the most extensive of these anomalies. For our purpose such a smoothing is desirable, since the local anomalies are due to the dis­ turbance of the field by magnetic materials near the surface of the earth and have no rele­ vance to the origin of the main field or of the secular variation. Although the smoothing is practically necessary and theoretically desirable, it does introduce an arbitrariness and uncertainty into the values at 10° intervals which have been chosen as representing the field. Thus in discussing the uncertainty of our results we have to consider not the experimental error, or the uncertainty in the reduction to epoch, but rather the uncertainty as to how far the figures in Vestine’s tables represent the smoothed version of the actual field that would be obtained from a close net of stations.
If the values from tables 1 to 6 used in the calculation of the drift were all independent it would be easy to determine its standard error. However, an examination of figures 1 to 6 shows that the non-dipole field retains the same sign over distances large compared to 10°, and that neighbouring entries in the table cannot therefore be considered as independent. The irrelevance of the actual number of tabular entries that are used may be seen by con­ sidering the effect of increasing their number by interpolation. The values at 5° intervals could be interpolated between the 10° ones and the number of values used in § 3 could thus be doubled, but as no new information is introduced, it is clear that the accuracy would not be increased. There is no entirely satisfactory way of dealing with this situation. We shall assume that the 36 tabular entries used for a given latitude are equivalent for the calculation of the standard error to N xindependent observations (tVj < 36) and shall then discuss the value of N v
In § 3 we have used equations of condition
and have chosen D to make He2 a minimum. For small shifts from the minimum we may
assume at each point
which gives e2 =
X\</>+D) = X '$ ) +DdX'/d(l>, X 2+ X '2+ (D d X 'W )2- 2XX - 2 + 2
where X has been written for X{<f>). Summing and differentiating gives
d{Xe2)jdD= 2DZ{dX'/d<}>)2- 2X [(X -X ') dX'/d^].
(1)
The minimum of Xe2therefore occurs at
Z[(X-X')dX'/d<t>] D ~
Z{dX Id
The change 8D in D produced by a small change 8X in one of the (X—X') is
WESTWARD DRIFT OF THE EARTH’S MAGNETIC FIELD
If all the ( X —X') change independently by amounts whose root mean square value is tr,
the standard deviation aDof D is
aD^ J { X 8 D 2)
JZ[8XdX'/d(l>Y Z{dX'ld<f))2
( 2)
5 if 8X and dX'\d<j> are independent.
Since from (1)
(2) may be written
=
Putting a2 = (Xe2)minJN (3) gives
J E { d X X I ^ ) 2V
d2(Xe2) / d D 2= 2 j2cr/J[d2(Ze2/) 2] .
Downloaded from https://royalsocietypublishing.org/ on 05 February 2024
==V[2(^2)min . / ^ 2( ^ 2) / ^ 2]
for the standard deviation of D assuming all the data to be independent.* If the A points used are equivalent to N x independent observations this expression must be multiplied by
(^e2)nun. and d2(Ze2)/dD2can be estimated from the values of at 10° intervals. The results are given in table 9.
T able 9
minimum he2 gauss2
d2(He2) 2 gauss2
lat.
80°N 60 40 20 0 20 40 60 80S
vertical
xlO -4 11 68
131 40 ' 84 85 121 166 39
east
xlO "4 10 20 21 20 22 22 42 52 76
north
XlO-4 12 30 16 61 21 21 69 37 48
vertical
xlO -6 25 198
318 170 199 122 250 235 36
east
xlO -6 54 148 135 82 88 74 89
103 142
north
XlO-6 47 57 11 54 40 85 40 28 92
There is no unique way of estimating the equivalent number of independent observations. In §3 we have used 324 values of each component. The number ofindependent observations of a given component cannot exceed this and is probably substantially less. A good, but far from perfect, representation of the field can be obtained by analyzing it into spherical harmonics of orders up to 6. This analysis requires forty-five constants. The number of independent observations must be substantially greater than this. We arbitrarily assume 100. An uncertainty of 50 % in this only affects the standard error by about 25 %, which is perhaps as good a result as can be expected; at any rate, the value obtained should not be wildly in error and is a good deal better than no estimate. The 100 independent observations are distributed among the circles of latitude in proportion to their length. The numbers are
lat. number,
80°
60°
40°
3-0
8-7
13-3
20°
0°
16-3
17-4
The 20, 40, 60 and 80° numbers occur twice, once in the northern hemisphere and once in the southern. When account is taken of this, they add to 100. The standard errors of the twenty-seven estimates of D computed in this way are given in table 8.
* This useful expression has been used previously to estimate the uncertainty in the determination of the thickness of the earth’s crust from gravity anomalies (Horsfield & Bullard 1937, p. 109).
82
E. C. BULLARD AND OTHERS ON THE
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The above argument assumes that the e’s are distributed at random, apart from the correlation between neighbouring values. Their actual distribution has been investigated by finding by interpolation the e’s for the vertical component of the field for every 10° of latitude and longitude when D has its optimum value of 6°-74. The results agree quite well with a normal law with standard deviation 0-016 gauss, the observed and calculated numbers
being:
residual (10-3 gauss) observed no.
calculated no.
—50 —40 —30 —20 —10 0 10 20 30 40 50
0 2 46 115 109 157 96 57 23 7
4 16 48
98 140 140 98 48 16 4
The table of e’s is not reproduced here as it does not show much of interest. The largest residuals (up to 0-048 gauss) are due to the change in shape of the 0-160 and 0-140 gauss curves in the neighbourhood of 50° S 40° E. The next largest are due to the elongation in an east west direction of the centre in southern Mongolia (see figures 1 and 2). Such changes cannot be compensated by any general shift of the isogams.
No systematic differences are apparent between the results for the three components, and their weighted mean is therefore taken. It is not clear if the results from the three com­ ponents are to be regarded as independent. They are derived from the same observations, and any one could be used to derive the other two by spherical harmonic analysis. Similarly it is uncertain how far the results from the different circles of latitude are independent. In calculating the standard errors in table 9 we have assumed that all the twenty-seven results can be regarded as independent. The weighted mean of the twenty-seven results is
D= 6°-74±0°-49 in 37-5 years.
The residuals from this mean are summarized in table 10. The table also gives their con­ tributions to %2. None of these is remarkable except that for 20° S. As the residuals from the east and north fields in this latitude are of opposite sign this does not appear to represent a systematic difference in the angular velocity of drift. If the twenty-seven residuals in table 10 are regarded as independent and to have the relative errors obtained above, the standard error may be calculated from their mutual consistency. The result for the mean D is
D== 6°-74±0°-55 in 37-5 years = 0°-180±0°-015 per year = (0-99 ±0-081) x 10~10radians/sec. = 20-0± 1-6km./year at the earth’s surface on the equator = 0-0344± 0-0028 cm./sec. at the surface of the core on the equator.
10. 1907-5 1945 T a b l e
R esid u a l s f r o m m e a n w e s t w a r d d r if t
to
lat.
80° N 60 40 20
0 20 40 60 80S
vertical
—1°-5 -2-5 -1 1
2-3 2-2 3-4 -3-9 0-0 7-2
residuals
_A.
east
l°-7 -2-0 -0-8
2-4 3-2 3-8 0-3 -2-9 40
north
l°-7 -2-5
5-2 -2-7
2-6 -5 1
0*2 -5-2
2-2
vertical
0-08 0-80 0-19 1-83 100 1-38 2-08 0-00 0-72
sum 8-08
X2
east
0-24 1-56 0-28 1-97 3-54 4-00 0-01 0-73 0-44
12-77 s%2 = 34.41
north
0-17 0-51 1-22 0-53 1-08 9-02 0-00 0-89 0-14
13-56
WESTWARD DRIFT OF THE EARTH’S MAGNETIC FIELD
83
The uncertainty, which is deduced from the agreement between the data for different components and latitudes, is close to that (±0°-49 in 37-5 years) found from the internal consistency of the results from the individual sets. This agreement is a valuable indication that the crude treatment of the interdependence of the data has not led to a gross error in estimating the uncertainty of the result.
The results given in the last column of table 8 and in table 10 show no change of angular velocity with latitude in excess of that to be expected from the uncertainty of the deter­ minations. The motion appears to be a uniform westward rotation superposed on local fluctuations.
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8 0 ° 6 0 ° 4 0 ° 2 0 ° O 2 0 ° 4 0 ° 6 0 ° BO*
latitude
N
F ig u r e 9. V a ria tio n w ith la titu d e o f th e w estw ard d rift betw een 1907-5 a n d 1945. (a) m ean ra te 6°-74 in 37-5 years, (b) constant lin ear velocity, (c) law o f ro tatio n o f sun spots a n d faculae; all have been m ade to agree at the equator.
The constancy of the angular velocity in different latitudes is of great interest. The results are illustrated in figure 9 which shows that they are inconsistent with a constant linear velocity. The law connecting the angular velocity of sun spots and faculae with their latitude is also shown, the error in the determination of the westward drift is too great to distinguish between this law and a constant angular velocity.
It is interesting to compare our result of 0T8°/year with Halley’s (1692, p. 571). He obtained 0-5°/year but remarks ‘the nice Determination of this and of several other par­ ticulars in the Magnetick System is reserved for remote Posterity’.
5. W e st w a r d d r if t o f t h e se c u l a r v a r ia t io n
Elsasser has remarked that Vestine’s maps giving the secular variation for 1942-5 and 1912-5 show a westward drift. This material can also be analyzed by the methods of §§3 and 4. As we are now dealing not with the changes in the field, but with changes in its rate
Vol. 243. A.
12
84
E. C. BULLARD AND OTHERS ON THE
ofchange, the accuracy is less than that obtainable from the non-dipole field. An examination of Vestine’s figures 124 to 135 (1947 a) shows that the isopors for the north component run predom inantly in an east-west direction and cannot be expected to give a good determination of the westward drift. Those for the east component are more favourably disposed. A deter­ mination could also be obtained from the vertical component but, in view of the large amount of work involved, we have confined attention to the east component. The results are given in table 11, where the standard errors are again calculated on the assumption that the data in the tables are equivalent to 100 independent observations.
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T a b l e 11. W e st w a r d d r if t o f ea st c o m po n e n t o f se c u l a r v a r ia t io n
latitude
drift
80° N 60 40 20 0 20 40 60 80S
12° 5 + 12°-2 0-3+ 3-9 3-2+ 3-8 8-3+ 4-0 12-8+ 4-6 15-3+ 3-5 15-8+ 4-8 15-3+ 6-1
18-7 ±10-8
m ean 9-56+ 1-6
m in im u m 2e2
(y/yr.)2
x 102 18 15 51
172 325 216 231 126 123
</2£ e 2
dD2
(y/yr.)2
X2
8
0-06
23
5-68
53
2-84
132
013
180
0-48
220
2-65
154
1-67
77
0-87
79
0-71
z%2 15-09
The results in high latitudes have, as would be expected, little weight and might have been omitted; it was, however, thought better to let them eliminate themselves from the result by their large standard errors than arbitrarily to exclude them.
All nine latitudes show a westerly drift. The values for 40° N and 60° N are low, but the errors are so large that there is no conclusive evidence for a real variation in angular velocity with latitude. The weighted mean result is
D x — 9°*6± 10,6 in 30 years.
X2for the nine latitudes is 15T which is a little high. It is possible that the secular variation data are equivalent to less independent observations than are those for the non-dipole field. If the standard error is calculated from the agreement of the results for the nine latitudes the
weighted mean is
^ = 9°-6±2°-0 in 30 years.
The difference between this result and that derived from the non-dipole field is
D l ~ D = 0°T4it:0o*069 per year.
It barely reaches twice its standard error. The decision whether this is to be regarded as significant rests almost entirely on the reliability of the standard error for the drift derived from the secular variation; in view of the uncertainty of this we do not consider the point as conclusively established. Six out of the nine determinations from the secular variation are greater than any of the twenty-seven determinations from the non-dipole field which suggests that there probably is a real difference. Such a difference is shown in § 7 to be possible theoretically.
WESTWARD DRIFT OF THE EARTH’S MAGNETIC FIELD
85
6. D r i f t o f t h e h a r m o n i c c o n s t i t u e n t s o f t h e f i e l d
In § 3 the westward motion of the field between 1907 and 1945 has been discussed by com­ paring directly the fields at these two epochs. It would be a tedious task to carry this work back to earlier periods; moreover, our knowledge of the field at earlier dates is so incomplete that it would be difficult to separate real changes from the uncertainties of interpolation.
There exists, however, another approach. Suppose the magnetic potential, G, to be analyzed in a series of spherical harmonics
Op
ZCnmP%{costf) cos
where 0 and (f) are the latitude and longitude, and Cnm and are constants. is related to the Gaussian constants g™and h™by
tan nuffi =
A steady westward drift of the field will be shown by a linear decrease of with time. The fifteen harmonics of orders 1, 2 and 3 give six different $ f’s. These have been calculated from Vestine’s data for 19.07*5 and 1945 and from eight other analyses for various dates going back to 1829 (Adams 1898, pp. 133 and 135; Chapman & Bartels 1940, p. 639; and Dyson & Furner 1923). The values of the $ f’s are given in table 12 and figure 10. Where the variation with time is reasonably linear a straight line has been fitted to the 9 points. The slopes of these lines are collected in table 13, which also gives the rates of change calculated from Vestine’s data only.
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T a b l e 12. L o n g it u d e s o f h a r m o n ic c o m po n e n t s
date
1829 1835 1845 1880 1885 1885 1907-5 1922 1945
author
dipole lat.
Erman-Petersen
78°-3
Gauss
77-8
Adams
78-7
Adams
78-4
Fritsche
78-6
Schmidt
78-7
Vestine
e 78-5
Dyson & Furner
78-4
Vestine
eal. 78-2
180° + <f>\
295°-3 296-4 295-7 291-9 292-2 290-5 290-1 290-9 290-0
n
179°-1 182-4 178-0 165-8 165-3 165-7 159-5 157-5 150-7
4
137°-8 135-4 134-2 123-8 122-2 123-2 112-6 105-1 99-1
4
208°-7 201-2 206-0 190-1 194-7 193-5 193-7 195-6 196*7
4
5°-7 8-7 4-2 -0-4 0-1 0-5 1-4 3-0 4-4
n
26°-6 31-4 24-3 19-8 18-9 19-8 13-0 5-9
1-3
180° has been added to 0} so as to give the longitude of the pole in the northern hemisphere.
T a b l e 13. W e s t w a r d d r if t o f h a r m o n ic c o m po n en t s
n
m
1
1
2
1
2
2
3
1
3
2
3
3
mean (« = 2 and 3)
non-dipole-§3 secular variation
westward drift °/yr.
1
1907-5 to 1945
\
1829 to 1945
0-003
0-062?
0-235 0-363 -0-080 -0-080 0-243
0-270 + 0-016 0-341+0-018 0-113? 0-037 0-234 ±0-024
0-136 0-180 + 0-015 0-320 + 0-067
0-199
12-2
86
E. C. BULLARD AND OTHERS ON THE
There is no evidence of any movement of the geomagnetic pole since 1880. Between 1829 and 1880 the observations suggest a westward shift of 3°-4. Such an angular shift in so high a latitude implies a linear movement of only 76 km. and, in view of the lack of movement shown by the later observations and the small amount of information available for the earlier analyses, it is considered that the observations during the last 120 years do not establish any certain shift of the geomagnetic pole. It is quite certain that its present west­ ward angular velocity is much less than that found for the non-dipole field in § 3. The pole also shows no perceptible motion in latitude (table 12 and figure 11).
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F ig u r e 10. Westward drift of the spherical harmonics.
F ig u r e 11. Longitude and latitude of the geomagnetic pole.
The two second-order harmonics and (j)\ show a marked westward drift of the same order as that previously found. <j>\ shows no appreciable drift and <j>\ none since 1880 (figure 10). Some irregularity is probably to be expected. The westward drift was found in § 3 to show considerable fluctuations from place to place. It is therefore to be expected that the rate of drift derived from the different harmonics will vary according to the region of the earth’s surface that has most weight in their determination. These variations might be expected to exceed the uncertainties in the determination of the drift of the individual harmonics, estimated from the scatter of the points in figure 11 from straight lines. Table 12 clearly shows this, as the variation between harmonics much exceeds that implied by the standard errors of the individual entries. Actually it is likely that the latter are themselves an underestimate, as many of the measurements have been used in more than one analysis.
In spite of these difficulties and uncertainties the examination of the harmonics is of value in showing that the results derived from a comparison of the fields in 1907 and 1945 is con­ sistent with the earlier results. The westward drift of the spherical harmonics has previously been noticed by Carlheim-Gyllenskold (1896). His perfectly genuine discovery has fallen into disrepute owing to his incorrect belief that the major part of the secular variation could
WESTWARD DRIFT OF THE EARTH’S MAGNETIC FIELD
87
be accounted for in this way, and to the absurd use of the idea by others to construct magnetic charts for remote periods. The secular change and the westward drift of the non-dipole field are related as are the changes in atmospheric pressure and the eastward motion of cyclones across the North Atlantic. The movement of the cyclones is genuine, but it does not imply the uniform motion of an unchanging system of isobars, or that the weather can be predicted far ahead. As a cyclone moves it changes in form and intensity and ultimately dies out. So it is with the centres of the non-dipole field; they move on the whole towards the west, changing in form as they go. They have not been observed over a long enough period to give a direct demonstration of their disappearance, but the continuance of the present rate of secular variation could build or destroy any of them in a hundred years.
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7. T h e ca u se o f t h e w e s t w a r d d r if t
In previous papers (Bullard 1948 and 1949
a,b), referred to as I, II a
the origin of the earth’s field has been proposed which will explain the existence of the west­
ward drift. The treatment of it given in II is, however, incomplete, and justifiable objections
have been made to it. *
It is supposed that the earth’s field is produced by a self exciting dynamo. The conductor
of the dynamo is the material of the liquid core of the earth, and its motion is a motion of
thermal convection produced by radioactive heating. The essence of thermal convection is
a radial motion, outwards in some areas and inwards in others. In a rotating sphere such
a motion is inconsistent with the conservation of angular momentum, and must necessarily
be combined with a radial variation of angular velocity, such that the material near the
outside of the core rotates with a lesser angular velocity than that inside.
The only tenable explanation that has been offered for the secular variation ascribes it to
the field produced by electromagnetic induction in material moving near the surface of the
core (Elsasser 1946; Bullard 1948). The rapidity of the change in field and the restricted size
of the centres of rapid change require that the motions should not be more than a few
hundred kilometres below the surface of the core; if they were deeper their effects would be
much reduced by screening by the overlying conducting material and would be more wide­
spread. It is natural to suppose that the non-dipole field is largely the integrated result of
the secular variation, and that its cause also lies at shallow depths in the core, though some
part may be of deeper origin. These relatively shallow features will move with the outer parts
of the core and will drift westward relative to the inner parts.
The above argument establishes that the minor features of the field and its secular varia­
tion may be expected to drift westward relative to the inner part of the core, but it does not
establish that they will have a westward drift relative to the solid mantle. At first sight there
are two possibilities, either the mantle is tightly coupled by viscous forces to the outer part
ofthe core, or it is not. Ifit is there will be no observable drift, and ifit is not the tidal decelera­
tion of the mantle will cause an eastward drift. This difficulty is removed by an examination
of the electromagnetic forces between the core and the mantle. Owing to the relatively low
conductivity of the silicates of the mantle the forces will be much less than those between
different parts of the core, but it turns out that, on any reasonable assumptions, they are far
* I am indebted to Sir Lawrence Bragg, M r T. Gold and D r W. Munk for their insistence that the dis­ cussion in my earlier paper was unsatisfactory. E.G.B.
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88
E. C. BULLARD AND OTHERS ON THE
larger than the viscous forces. The electromagnetic forces differ from the viscous ones in
providing a coupling not merely with the outer part of the core but with the core as a whole;
they therefore cause the mantle to follow not the outer part of the core, but some weighted
average of the whole core. Such a coupling therefore allows the outer part of the core, and
with it the non-dipole field and the secular variation, to drift westward relative to the mantle.
Calculations are in progress to find the field produced by specified motions in the core.
From these it will be possible to compute the forces on a mantle of given conductivity. Until
these computations are complete we must be content with cruder arguments. Let it be
supposed that the core is divided into inner and outer parts of radii and a and that each
rotates like a rigid body. Let their angular velocities be o)b and (oa and their conductivity k.
Suppose them surrounded by a mantle of outer radius R and conductivity /q rotating with
angular velocity oj1. This system departs from reality in that the continuous radial variation
in angular velocity in the core is replaced by a discontinuous one. Further, no radial motion
is provided, and thus the system cannot act as a self-maintaining dynamo; we therefore
arbitrarily suppose a field to exist and compute the couple on the mantle. There is some
latitude in the choice of field. We take a uniform field parallel to the axis and a central
dipole as representing two extremes in radial variation between which the truth must lie.
In fact the results do not depend critically on the radial variation of the field so long as it
has its known value of about 4 gauss in the mantle near the core.
The relative rotation of the two parts of the core produces currents currents in Elsasser’s
notation) flowing in meridian planes downwards near the equatorial plane thence towards
the poles along paths near the axis and back towards the equator near the surface. These
currents produce a toroidal field
(T2)which encircles the axis from west t
hemisphere and from east to west in the southern. All this has been thoroughly discussed in
II and III. If the mantle is a conductor of electricity the S2 currents on their way to the
equator will flow partly in the mantle and their interaction with the dipole field there will
produce a couple. We require to know for what angular velocity of the mantle this couple
will vanish, and how long it will take it to re-establish this angular velocity if it is disturbed.
The electromagnetic problem can be solved by the methods of III; the details of the solu­
tion will not be given here as it is lengthy and uninteresting. The couple T is found to be
given by
for a constant inducing field H0, and by
r
=
£-vK,a*Hlt(ab-6t.)
45 1 ou b *'
W 1
-ti.+ 1
ti"JSr+-.«1i,/.«a3+/*(Vl-. kJ
k)
a ’j Rh
for a dipole inducing field giving a field at The couple vanishes if
0, r =
o)l = ( b55/a)
o)b(-\1 —
b5/a5)o)a for the constant field
or
o)l — ( b22/a)
(t)b+(1 —
b2/a2)o)a for the dipole.
If the core is divided into two parts of equal volume this gives
(ol — 0-32^ + 0-68^ for the constant field
or
o)x= 0-63cq, + 0-37wa for the dipole.
WESTWARD DRIFT OF THE EARTH’S MAGNETIC FIELD
89
The angular velocity of the westward motion of the surface of the core relative to the mantle
would therefore be
(ol —(oa— 0'32((ob—(Da) for the constant field
or
o)l —(oa= 0‘63(o)b—o)a) for the dipole.
In practice, the constant will probably lie between these two values and for the present purpose we may take as a rough approximation
6>i-<ya = 0*5(<y6-<ya).
(4)
If the relative angular velocity of the outer part of the core and the mantle is to have the value found in § 3, o)b—(t)a must be about 0-36° per year. The maximum relative velocity at the boundary of the inner and outer parts of the core will then be 0-055 cm./sec. The argu­ ments of II (p. 444) would give the corresponding radial velocity as 3-4 x 10-4 cm./sec. This may be somewhat underestimated as it is computed on the assumption that angular momentum must be conserved in the radial motion of every particle, in fact the electro­ magnetic forces tend to restore a rigid body rotation and thus increase the radial velocity necessary to produce a given differential rotation.
The maximum toroidal field is shown in III to be
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whereHx is the field at
r — b(either uniform or dipole). The above value of o)b—(oa gives
150 gauss for the toroidal field if
k= 3 x 10~6, = 4 gauss and the core is divide
equal parts.
The above explanation of the westward drift is imperfect in that the field in the core is
assumed instead of computed from the equations of magneto-hydrodynamics. Eventually
it should be possible to remedy this deficiency, but it is unlikely that the order of magnitude
of the results will be changed. The couple between the mantle and the core is due to the
interaction of the dipole field in it with the small part of the S2 current that leaks from the
core to the mantle. This is approximately represented in the above treatment, and the main
elements of the field which have been omitted (the Xf fields and the *£§currents of II) produce
no couple.
The artificial substitution of a discontinuous for a continuous radial variation in angular
velocity within the core has removed the possibility of features due to eddies at different
depths moving westward with different velocities. The causes of the secular variation must
lie within a few hundred kilometres of the surface of the core or their changes would be
screened by the overlying conductor, but there is no reason why the non-dipole field should
not possess a more permanent part whose causes lie rather deeper. In this way it seems
possible to account for the westward velocity found in §5 for the secular variation being
greater than that found in §3 for the non-dipole field.
The dipole field is believed to be due to motions extending through the major part of the
core. It therefore seems reasonable that it should partake of the average motion of the whole
core and not of that of the outer part only. This is in agreement with observation, but a
detailed treatment would require a more thorough discussion of the motion of the terrestrial
dynamo than can be given at present.
90
E. C. BULLARD AND OTHERS ON THE
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If the core does not rotate relative to the mantle with the angular velocity found above,
a couple will be produced tending to restore that rate of rotation. With a uniform field the
difference from the equilibrium rate will be reduced to 1 in a time r given by
45 ^-\-k1/k —(k1/k —1) a5fR 5
T~ 8
(i-0 7 .R 5) (1 /4 + 1 //,) ’
where Ic and Imare the moments of inertia of the core and the mantle. The conductivity of
the mantle of hot silicate is only vaguely known. A study of induction by fields of external
origin shows that at a depth o f600 km. below the surface of the earth the conductivity exceeds
10_11e.m.u. If it exceeded 10~8e.m.u. the secular variation would be greatly reduced by
screening (I, p. 256). Extrapolation of the experimental and theoretical work of Coster
(1948) suggests that the conductivity rises to about 3 x 10~9e.m.u. at a depth of 1000 km.
and then falls to about 10-11 e.m.u. near the core. A conductivity of 10-1°e.m.u. will be
adopted; this corresponds to a resistivity of 10 ohm cm. Since k is about 3 x 10~6e.m.u.
kJ k <41. The terms in
a5/R 5and in / c//mcan also safely be dropped leaving a time
of r —2
D/k1H%,where D is the density of the core. With D = 10-7 g./cm.3, H0 — 4 gauss,
this gives 1010sec. or 300 years. A similar result is obtained if the couple appropriate to a dipole
inducing field is used.
If the tidal deceleration of the mantle is (bl radian/sec.2it may be shown that the westward
drift will be reduced by WjT, With r = 1 0 10sec. and
2 x 10~22 sec.-2 this gives
2 x 10-12sec.-1 (0-004°/year) which is negligible. The coupling between the mantle and the
core is therefore sufficiently tight to prevent the tidal deceleration of the earth from appreciably
reducing the westward drift. If the conductivity of the mantle were a hundred times less than
we have assumed, this would no longer be true; the specific resistance of the mantle would
then be over 1000 ohm cm., which is certainly too high.
If the mantle rotates with the angular velocity given by (4) no current flows in it. If its
angular velocity departs from this by an amount 8oj the maximum value of the current
density is approximately aH08(o. If
80)were such as completely
drift, that is if the mantle moved with the outer part of the core, this would give 1-4 x 10“12
amp./cm.2 in the mantle. This is small compared to the currents in the core which are
about %KbH0((i)a—G)b) or approximately k/k1 times as great.
It might be suggested that the observed fluctuations in the rate of rotation of the earth
are due to electromagnetic coupling of the mantle to turbulent motions in the core. The
above results show that this cannot be so, since the time constant of such processes will be
at least as great as the 300 years calculated above, whilst the observed changes sometimes
occur in a few years.
It remains to show that the neglect of viscous forces is justified. If the mantle starts to
rotate relative to the outside of the core with angular velocity <y, a boundary layer will be
formed in which the velocity changes from that of the mantle to that of the core. This boun­
dary layer will at first be thin compared to the radius of the core. Let its thickness be s.
The viscous couple between the mantle and the core will be TT2(or]a?ls, where 7 is the viscosity.
In order to supply this couple from electromagnetic forces the relative angular velocity of
mantle and core must depart from the value given by (4) by an amount 8(0 given by
8(1) _ 10/77/
WESTWARD DRIFT OF THE EARTH’S MAGNETIC FIELD
91
Putting rj = 10-2g./cm.sec., /q
—10~10e.m.u., a x 108cm. and 4 gauss this gives
8o) 0-06
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The viscous forces are therefore powerless to alter the westward drift appreciably even if
the boundary layer is as thin as one centimetre. The same conclusion may be reached by
considering the time required for viscous forces alone to reduce the relative angular velocity
ofcore and mantle to
1/eofits initial value. This time is approximately sDa/5rj or 2 x 103jyears
(j-in cm.). For any reasonable value of s this is much greater than the corresponding time
for the electromagnetic forces, and for very moderate values of s may exceed the age of
the earth.
From the above discussion it appears that the dynamo theory requires a westward drift
of the non-dipole part of the earth’s magnetic field and of the centres of secular variation,
and that the observed rate of drift can be accounted for by reasonable values of the quantities
concerned. The westward drift gives the most direct estimate of the velocities in the core, and
it is encouraging that the result obtained is of the same order of magnitude as that deduced
from the secular variation (II, figure 2). An independent estimate can be obtained by
computing the critical velocity necessary for self excitation of the dynamo. An investigation
of this is in progress.
Whilst the above explanation of the westward drift grew out of the dynamo theory, it
would also be consistent with other theories of the origin of the main field, provided the
differential rotation of the core occurs. The explanation uses most of the mechanism of the
dynamo, but does not exclude the possibility that, in spite of the existence of this mechanism,
the field might be produced by some other means.
Elsasser (1947, 1949) has suggested that the westward drift is to be explained by a slowing
down of the rotation of the core by tidal forces. Reasons for supposing this effect to be
negligible have been given in II, p. 438. K. Runcorn and W. Munk have made unpublished
suggestions that the westward drift is due to a westward motion of eddies in the core through
the general body of fluid. This idea is worthy of detailed investigation; in particular it would
be interesting to know what variation of drift with latitude would be expected.
The computations on which this work is based were done in the Computation Centre of the University ofToronto with the help offunds supplied by the Canadian National Research Council and the Defence Research Board and in the Scripps Institution with funds supplied by the United States Office of Naval Research. We wish to express our thanks to these bodies for their assistance.
R eferences
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Rep.Brit. pAs. p. 109—136.
B ullard, E. G. 1948 Mon. Not. R . Astr. Soc. Geophys. Suppl. 248—257.
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B ullard, E. C. 1949 b Proc. Roy. Soc. A, 199, 413—443.
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C h ap m an , S. & Bartels, J . 1940 Geomagnetism, vol. 2. O xford: C larendon Press.
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92
E. C. BULLARD AND OTHERS
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Halley, E. 1692 Phil. Trans. 17, 563-578.
Horsfield, W. & Bullard, E. C. 1937 Mon. Not. R. Astr. Soc. Geophys. Suppl. 4, 94-113.
Vestine, E. H., Laporte, L., Cooper, C., Lange, I. & Hendrix, W. C. 1947 Description of the earth's
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change, 1905-1945. Washington: Carneg
no. 578.
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analysis. Washington: Carnegie Institution, Publ. no. 580.
Downloaded from https://royalsocietypublishing.org/ on 05 February 2024
Downloaded from https://royalsocietypublishing.org/ on 05 February 2024
PRINTED IN GREAT BRITAIN FOR THE ROYAL SOCIETY AT THE UNIVERSITY PRESS, CAMBRIDGE (BROOKE CRUTCHLEY, UNIVERSITY PRINTER)