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Some morphological aspects of geomagnetic secular variation
Planet.
Space Sci. 1965.
Vol. 13.
pp.1097
to 1110.
Pergamon
Press Ltd.
Printed
in Northern
Ireland
SOME MORPHOLOGKAL ASPECTS OF GEOMAGNETIC SECULAR VARIATION Department
L. SLAUCITAJS* and 33. E. WINCH of Applied Mathematics, Unive~i~ of Sydney, Sydney, N.S.W., Australia (Received 9 June 1965)
Abstract-Analysis of geomagnetic secular variation data, in the t&e elements X, Y and Z at five observatories distributed over the Earth, shows that amplitudes obtained for a 50 year cycle by harmonic analysis are highly significant. Amplitudes of higher order harmonics are found to be either not as significant or not significant, indeed, the cycle can be seen dire&y on the graphs of ob~ational data. These results are confirmed by power spectrum analysis. Using phase angles obtained for the 50 year (semicentennial) cycle from different stretches of data, an improved value for the period is found to be 61 f 6 years. The approximate semicentemkl values are interesting in view of the theoretical possibility of an oscillation of an S,O field of period 77 years superimposed on the main field. INTRODUCTION
of the author@ indicated a “pulse” or “period” of appro~mately 50 years in secular variation of the geomagnetic field. This was confirmed by independent papers of Barta@“) and more recently by Gaibar-PuertaC-‘1 and Slaucitajs.(8) Wasserfall,(*) using data for Oslo 1820 to 1948 mentions some “nutational” period of 70 years, Slaucitajs(l”) made a harmonic analysis of 44 years of secular variation data from a South American station, so that a fourth harmonic represented the effect of the solar sunspot cycle. Stoyko@l) indicates that the fluctuation in terrestrial time in a period of about 50 years is correlated with secular change in the total intensity, and Elsasser(12) comments that periods of 40 years are fairly representative of the secular variation. For the IAGA Meeting in Toronto, 1957, a communication on forecasting of secular variation was presented by Kalinin(13) in which the author mentions that part of the changes in the geomagnetic field effected by internal causes has a quasi~riodical character at observatories in Eurasia with a period of a few decades and that another part has an eleven year period connected with the geomagnetic activity. Yukutake(14) indicates that analysis for short intervals of secular variation data reveals a 50-year and 25-year period, Depietri(15) analysing secular variation for the Northern Hemisphere between 350” and 80”E received a period of 55 years. Recently Degaonkar(16) analysed AH, AZ and AF for three American stations with respect to the 11 year sunspot cycle, and Yukutake’s paperQ7) on the eleven year period variation in the geomagnetic field associated with the solar activity was presented to the IAGA Symposium in Pittsburgh. These researches are largely empirical, and one must be cautious since(l@ periodicities found by harmonic analysis and not predicted by previous theoretical considerations may be untrustworthy, as many complications are capable of giving spurious periods. As we know Bartels’19) indicated the predominantly local character of secular variation and after McNish(20) it is preferred to relate the major part of secular variation with the residual, non-dipole field. Nevertheless the morphology of the phenomenon shows that some periodicities in secular variation exist(21)and one of these is the approximate semicentennial. One
* On leave from the National University of La Plats, Argentina. 1097
L, SLAUCTFAJS and D. E. WINCH
1098
The physical theories of the origin of the main field and secular variation field based on magnetohydrodynamics need to meet the morphological aspects shown by and deduced from observational data. Now, Nagata and Rikitake(22) indicate the theoretical possibility of an oscillation of an S,Ofield about the main field with a period of 77 years. Although one could not conclude that the observed cyclic changes in AX, AY and AZ (see below) are directly TABLE 1
observatory
Geographic Long. Lat.
1. Wyssokaja Doubrawa
56” 44’N
61” 4’E
2. Niemegk
52” 4’N
12’ 41’E
3. Pilar
31” 40’S
296” 7%
4. Cheltenham
38” 44’N
283” 1O’E
5. Kakioka
36” 14’N
140” 11’E
FIG. 1. GEOMAGNETIC COORDINATESOF OBSERVATORIES WHOSE SECULAR VARIATION DATA ARE U~~~A~AL~S.
related to this quadrupole oscillation suggested by the theory, it is most interesting to find secular changes with a period comparable to the theoretically suggested one. It is the purpose of this paper to present harmonic and power spectrum analyses of the armual mean rate of change of the elements X, Y and 2, denoted AX, AY and AZ using all available data at the magnetic observatories listed in Table 1. Figure 1 illustrates the distribution of these observatories over the Earth. In the well-known graph of Declination and Inclination for London from the 16th century, now expanded till mid-20th century by Gaibar-Fuertas(6) in which short period features are not obliterated to give prominence to an apparent 500 year cycle, a superimposed cycle of se~centenni~ period is also suggested.
SOME MORP~OL~IC~
ASPECTS OF GEO~GNE~C COMPUTATIONAL
SECULAR V~A~ON
1099
DETAILS
First the graphs of observed values of AX, AY and AZ and of running eleven year means to eliminate the solar activity period were prepared. These graphs suggested (as already reported by Slaucitajs(l)) testing the period mathematically. The secular variation corresponds to first differences of the main field values, and so does not contain any linear trend in the latter, so that any superimposed oscillations are more readily seen. For intervals of data longer than about 70 years, any parabolic and cubic trends in the main field result in linear and parabolic trends respectively in the secular variation field which distort any oscillations. For this reason, harmonic analysis of xxx&r variation values over 100 years to test the hypothesis of a 50 year cycle will not lead to significant results. An attempt to fit by the Method of Least Squares the regression model 4
Y = a0 +kslak sin mkt + $ bKcos mkt k=l
where m = 27~150to given dataye, at equally spaced intervals t = 0, 1,2, . . , , N to the normal equations
(1) 1 leads
If now N is taken to be 50,100,150, . . . then these normal equations reduce to diagonal form, and estimates of a, and bBwill be entirely uncorrelated; from a statistical viewpoint this is very desirable. In each analysis N has been taken as 50. The solution of the normal equations is now
2iv-1
a, = -
yr sin mkt NC t-0
b, = ;;i;yt
cos mkt
k = 1,2,3,4
k = 1,2,3,4.
A computing algorithm given by Goertzel WI) takes one tenth of the computer time required for direct evaluation of a, and bk using sine and cosine subroutines and this has been as follows: with
ET,=0
1100
L. SLAUCITAJS
and D. E. WINCH
compute for k=N-
1, N-2
N
-z bk = yO f N 3ak
=
,...,
I,
V, cos mk - U, U, sinmk
Since the sum of squares due to regression is advaO + 2 adad2)
+ 2 bk(bkN/2)
hence the residual sum of squares is
where .r2 is an estimate of the variance around the regression relationship. Confidence limits for a, are a,, f tg l/N and for a, and bKare ak f 6 and b, f 6 where 8 = t&2/N),(“) where t, is found in Student’s t table with N - 9 degrees of freedom.(26) The expression (1) may be put in the form
a, + i Cksintmkt + #kl k=l
Noting that annual mean values given for the years x and x + 1 have a first difference corresponding to the secular variation at epoch x + 05, it proved convenient to refer all phase angles to epoch 1900.5. From confidence limits a, rt: 6 and bk f 6 for a, and 4, confidence limits for C, and &.k,confidence limits for C, and & are given by Whittaker and RobinsorP) as C, f 8 and 46, f s/C,. The quantity s/c& will be in radian measure, The results of this analysis are given in Table 2. For those cases in which the 50 year cycle is s~g~fi~nt and higher harmonics not as sibilant, or insig~ficant, the following theory to indicate a more suitable value for the period of the cyole is applicable. The problem is to fit the curve y=Asinkt+Bcoskt = C sin (kt + a) where k = 24T and A + iB= Cexpia
(2)
SOME MORPHOLOGICAL
ASPECTS OF GEOMAGNETIC
SECULAR VARIATION
1101
to data which is really of the form y = D sin (k’t + p) where k’ = 277/T’
111 0
E g
20
WYSSOKAJA
10 1
AX
DOUBRAWA
I
L
-50
s. 10 c. 0E 0 ‘D > 20
20 10 0 -10 E -20
, 1990.5
I
v
I
1900.5
1910.5
I
I
1920.5 Epoch,
1930.5
1
I
1940.5
19505
I
19605
yr
FIG. 2. WYSSOKAJA DOUBRAWA. UNRROKENLMEISSECULARVARIATION,DASHED LINRISREOREWON CURv@FORFIRSTFWTYYEARS,DO~DLINEISREGRESSIONCURvEFORLASTFIFNYEARS.
Solving the least squares equations for A and B, and substituting into (2) gives C exp ia = s [ (exp i[p + (N -
-
l)(k -
{exp -U
k’)/2]} zz Nii:
+ (N -
1 $
l)(k + k’)P])
if
Nti
I::$]
Assuming k w k’, the first term of the expression dominates, and hence a M B + (N - l)(k’ - k)/2
(3)
If a second analysis is performed on a further set of observations, whose tist observation is the mth after the fist observation of the fist set, and a, is obtained as the phase angle referred to the point t = 0, then a2 + km = /? + k’m + (N -
l)(k’ - k)/2
(4)
L. SLAUCITAJS
1102
and D. E. WINCH
Solving (3) and (4) for k’, the expression for T’ is obtained: T’ = 360mT/{(cc, - a)T + 36Om))
(5)
where as and a are in degrees of arc, and T is 50. If confidence limits for a and a2 are a & 6, and a2 f 6, respectively, then confidence limits for T’ are given approximately by T’ f. d(d,Z + 6,2)T’2/(360m) NIEMEGK
70
60 50 g
40
:
30
sl :
.
20 10
;; b 2.
50 -
i
403
0 m
AZ f
30 zo-
\
lo-
, \
O-10 -20
-
-30 -40
-
-50
-
-60
-
-70 -90
-
-90 -
f 1990-s
1
1900 5
I
1910.5
I
Epoch,
FIG.
3.
I
1930.5
1920.5
I
1940.5
t
1950.5
I
1960.5
yr
NIEMEOK. UNBR~KENLINEISS@CULARVARUTION,DASHED LINEIS REGRESSIONCURVE FORFIRSTFlFTYYEARS,DO-fTEDLINEISRRGRESSIONCURVEFORLASTmFIYYEARS.
The results of this analysis are included in Table 2, except for observatory 3, Pilar, where m was small and the results were non-significant. Although 95 per cent cotidence limits are given for amplitudes and phase angles, confidence limits for the improved period are approximately 60 per cent, due to the nature of the approximation made in the theory. Twenty-five autocorrelation coefficients were computed for the power spectrum analysis, since this is the smallest number which would give a spectral peak corresponding to a
SOME MORPHOLOGICAL
ASPECTS OF GEO~AG~E~C
SECULAR
VARIATION
fifty year cycle. The cosine transformation(a7) of the autocorrelation computed using the Goertzel algorithm described above. SOURCES
1103
coefficients was
OF DATA
The principal sources were the Bock-Schumann catalogue@‘) and the Nagata-Sawada catalogue,@) Annual mean values in the interval not covered by these two catalogues,
6
-60
E -70 E -80 ::
5 Ti -50 tj - -60 > -70 s 2 x
*
-so 40 30 20
.
10 0 -10 -20 1910*5
t920*5
19305
1940 5
1950~5
19606
Epoch. yr
Fm. 4. PILAR. UNBROK.ENLBIBI~SECULARVARIATM)N,DASHEDLINEISREGRFS~IONCUR~EFOR FIRST FWIT YEARS,DOTTED LINE IS REGRESSION CURVE FOR LAST FIFTY YEARS.
1949 to 1955, were obtained from yearbooks of the observatory concerned, or from the institute controlling the observatory. Mr. B. R. Leaton very kindly supplied missing values for Wyssokaja Doubrawa. CONTUSIONS
Power spectrums and harmonic analyses indicate that the amplitude of a 50 year cycle is highly sign&ant at the five observatories distributed over the Earth, except for AY and AZ at Kakioka; in the case. of AZ this seems to be due to the great variability of the earlier values. The amplitudes of the higher order harmonics are either not as significant or not significant. The average result for the improved values of the periods is found to be 61 ,j, 6 years(60 per cent confidence limits). This result taken with the theoretical prediction of an oscillation of an axial quadrupole field having a period of 77 years, suggests that spherical harmonic coefficients of the main field should be computed at at least two year intervals, using all the data contained in the
106f 3 218ir 14 156i: 9 12Of7 80 f 17
255rfr:8
15 k 1
26 & 6
27 i 4
20 i 2 11 f3
21 + 1 16 &2
20 f4
40 l 3
O&2 -312
44 f 1
47 a 2
AY 18905-1939~5
1912.5-1961.5
14 f 4 12 f 4
16OrtlO 336f 15 324& 16
20 f 3
16 f4
16 + 4
312
-50 xt 3
-52&3
1908%1957.5
3. Ax 1905.5-19545
8zt9 10 f 3
221f 17
29 + 9
8k6
19125-1961.5
411 6f2
10 f 2 63~3
6f4
13~6
2&l
l&l
6&3 13 + 5
G P
AZ:1890*5-1939.5
19125-1961~5
2. AX 1890+-1939.5
1911*5-1960.5
AZ;1887.5-1936.5
1911+-1960-5
279f 3
105f 3
21 & 1
2&l
-3 + 1
AY 1887S1936~5
142i 5 106$: 10
lPll+-19605
1. AX 18875-1936.5
29 It 5
41 deg.
32 & 3
c, li
-321lz4
a0 li
lztl 3f2 6rtr9 5&3
301+ 14 148f 22 308f 63 343i 20
295+ 21
514
5zt4
4xt3
16 f 32
306f 17
3rtr2
143i 14
2&6 7f4
70 i 40
201f 55
240f 54
338f 44
88 & 75
101+ 37
274f 61
326+ 45
222f 46
350i 34
115f 217
120rt 22
298i. 17
3+1 2331
359i 65 318i 31
4s deg.
2f3 IO & 5
G P
31 f 375
132It 26
2134 44
11 f23
147i 27
#e deg.
2414 44
7f5
5+6
5*4 354
2rt3
4It9
3f2
3&l
413
3&2
4&4
234& 51 201i_ 93
195f 113
138f 117
58 f 41
125i: 19
166& 50
225f 41
109i-66
179f 67
109zt86 134f 17
288& 51
3f3
l&l 3ztl
164 deg.
81 f 16
59 zt 4
Improved period, yr
~ISPIL~,OBSERVATORY
G ?
OBSERVATORY~~SWYSS~KAJADOUBRAWA,ORSERVATORY~ISNIEMEGK,OBSERVATORY 4IS ~L~~M,O~~VA~RYsIS~OKA
-38f2
Ob. El. hterval
INDICATED ~ERTHEHIWXN~‘?NTERVAL”.
TABLE2. NINETY-FIVE PERcENTCONmDENCEL~sFORAMPLITUDESINGAMMASPERYEARANDPHASEANGLESINDEGREESGIVENATEPOCH 1900.5 OPFOUR HARMONIcSOFA%YEARFUNDAMENTAL. %XTYPERCENTCONFIDENCELMTSFORTHElMPROVEDPERIOD. THEFIFTYYEARINTERVALSIJSEDFORANALYSISARE
? $ h
r
-57 ri: 1
1908s1957.5
-20 f 5
-9 f 1 -9 i 1
1911.5-1960.5
AY 1902.5-1951.5
-17$:2
AY 1897.5-1946.5
19W5-1958.5
AZ 18975-19465
4 j, 15 -2 rir13
-16 & 1
7&4
1909.5-1958.5
6&6
1909.5-1958.5
-53 xk5
1911.5-19605
5. AX 1897.5-1946.5
-56 rt 5
AZ 19025-19515
1911*5-1960.5
-37 ic 4
4. AX 1902.5-1951.5
AZ 1.905*5-1954.5 -12zk2 19065-1955~5 -12 * 2
-58 zk1
hY 19055-1954.5
22 -+ 18
18 i22
l&2
4f2
15 f 6
13 zk8
46 f 7
44 + 7
10 i 2
10 It 2
53 f 7
40 f 5
24 c?c 3 23 f 3
13 12
13 & 1 l&l
171j, 47
142& 70
356& 169
240It 35
115* 21
119& 36
120-+:8
125+9
171& 20
174& 9
92 f 7
129rt 8
5 i 22 6 It 18
4rrt2
i&2
5&6
3f8
41t7
517
4rt2
5*2
19 f 7
24 f 5
3&3 2xt3
346f 7
2&2
349Jr 7
1391:7
143$5
18 f 234 144i 172
24 z!c29
109z!x194
112f 68
143f 160
145&- 82 96 &9l
227f 23
218& 20
135f 12 59 & 21
343& 93
357zt 59
350f 55
355f 110
3 i 22 8 -4r18
3f2
2rf2
2f6
J&t?
7&7 10 & 7
3f2
4f2
4rt;5 17 St 7
6rf3
6+3
2-rt2
21-l
79 Ii=31
l&l
167rir57
326f 376 100+ 133
10 + 46
127f 66
244 & 174
241 f 93
358f 40
2 i 61
93 f 35
96 i 26
318rt 23
88 f 83
318zk66 318rt 22 212It 64 299* 102
19 $I 22 10 & 18
31 rfr:77
4f6 2&2 5&2
333xk491
lct8
2614 64
244f 77
5&7 6f7
139f 58
114f 52
212 2f2
234I: 31
5kt5 13 rfl:7
167&56
239f 34 225I 36
5f3 5k3
201+ 52
229f 26
212
240+ 29
55 f 59
37 f 27
21 & 18
53 ct 27
55 f 11
53 zk 12
117f 45
1106
L. SLAUCITAJS and D. E. WINCH
50
CHELTENHAM AX
30 20 10 40 0E -10 ~ \ -20 \ \
-30 -40 -50 -60
[
; -70 E -80 ~ 8l
.
10 i 0 ;; - -10 & > -20
AY 1
-90 -100 -110 -120 -130I__-L_i_-_. 1900.5 1910.5
1920.5
1930-S
1
I
I
1940.5
1950.5
1960.5
Epoch, yr FIG.~.~IELTDIHAM.
UNBROKEN LINE IS SECULARVARIATION,DASHEDLINEIS REGRESSIONCURVEFOR~TmFTYYEARS,DO~D~ISREORESSIONCURVB FOR LAST FIFTYYEARS.
catalogues mentioned above, and examined for superimposed periodicities. Least Squares applied to a model of the form
y =
2 a*xf
i-o
The Method of
+ anfl cos (27rxfT) 4 aS.+zsin (2427
where ordinates y, are the spherical harmonic coe5cients of the same kind, x is the year, and T is the period, would be appropriate. From the paper by Nagata and Rikitake (s2)the period of oscillation of the SBofield is deduced as 23ra 1 75-p T=zg J( 3 )
SOME MORPHOLOGICAL
I
_---__L*;--5 r’ 0” 4r
P
.:
1 . . . . . L ,.................:
l_______,____ix.I~z
SECULAR VARIATION
,____.i
:r-_--i...
i I : I :
ASPECTS OF GEOMAGNETIC
:..... .. .
I
,.......,.....................
-...
?
ci
i
I
’
I
ii, “0
I
I
“0 Ad~/~(sourwo6)~X~~suep
0
JaMOd
I
5:
1107
-
LA
__
: :.
Period.
56
,:
I
2.9
I.,
Ii
II,31::fL
Ll
i.f
:.!
. . . :: i
.
. . .
i
yr
:
1.: . ..:
:
’
2.0
I
I
I
;
I
L ,
,I;;
i
: : : :
:” . . . : : :
I
Ir’ I:
:: :: :: :.;
i”
1
I.
I
:.
FIG. 8. NIEMEGK. UPPER 90% CONFIDENCE ~rmrf? ARE SHOWN FOR EACH ELEMENT.
A12.5i
A-: : : : : : . . : : : : : : : : . : : : : : : :.:
: :
I:
-1
I....
Fwiod.
yr
’
(
i
LIMITSARESHOWNFOREACHELEMENT.
FIG.9. PILAR. UPPER 90% CON~DENCE
_r?
AV .............A2
AV ,...,........AZ
3 Ax
Spectrum ________
Spectrum 2 ______ -_ A)(
5 m
c
SOME MORPHOLOGICAL
ASPECTS OF GEOMAGNETIC
SECULAR VARIATION
yz
. ..... ..... ............ ... r______-_-____ ___ ..:
....:
,__-__J
:......
X>N
3
: .. . .. . .
-ada
L____,
:*.: .. . . . .. ,. .
. .. .... . :..
.
L-_______,
r_-___, r_____J
: . . . . .. ..
I :
L_
I
.i i
L____?
,-_______J
r
rd
: . . . . . . . . ra . . . .
,
:......... ...:
I
‘6
-6
Ida /z(
SDWWD~ )‘Xa!suap
P Jarnod
1109
1110
L. SLAUCITAJS
and D. E. WINCH
where a is the radius of the core, p the density of the core, S, a steady poloidal field in the core. When T = 50years, the value for S, is found to be 2.8 e.m.u., a not unreasonable value. REFERENCES 1. L. S~~vcrr~a, Contri6. Baltic Univ. 63, 1 (1948). 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
A Magyar Tudomdnyos Akademia Miiszaki Tudomdnyok Oszttit’ya Kozlemenyei V, 1 (1952). BARTA,Magyar Allami ZWvi2s Lorand Geofzikai Zntezet Geojizikai Kiizlemenyek III, 1 (1954). BARTA,Acta Geol. IV, Fast. 1 (1956). GAIBAR-P~ERTAS, Observatorio de1 Ebro. Mem. 11,201(1953). GAIBAR-P~ERTAS, Geof. pura e appl. 29,22 (1954). GAIBAR--PUERTAS, Publ. corn. 50” aniv. Obs. Magn. San Miguel, Lisboa (1962). L. SLAUC~~AJS, Pure Appl. Geophys. 59/111 75 (1964). K. F. WASSERFALL, J. Geophys. Res. 55,292 (1950). L. SLAUCITAJS, Publ. de1 Observ. Astron. de la Plats, Ser. Geof VII, 1 (1951). N. STOYKO, C.R., Acad. Sci., Paris 234,1798 (1952). W. M. ELSASSER, Amer. J. Phys. 24,87 (1955). G. G. G. C. C. C.
BARTA,
12. 13. Y. D. KALm, Trans. Toronto Meeting, ZAGA Bull. 16,368 (1960). 14. T. YUKUTAKE,J.Geomag. Geoelect. XII, 102 (1961). 15. C. DEPIETRI,Geof. e Meteorol. VIII, 5/6 (1960). 16. S. S. DEGAONKAR, J. Geophys. Res. 68,6206 (1963). 17. T. YUKATAKE,ZAGA Symp. on Magnetism of Earth’s Interior, Pittsburgh (1964). 18. H. JEFPREYS and B. S. JEFFREYS, Methods of Mathematical Physics, page 451. Cambridge University Press (1963). 19. J. BARTELS, Preuss. Meteorol. Inst. AbhandZ. 8, (2) 23 (Veroff.Nr. 332) (1925). 20. A. G. MCNISH, Trans. Amer. Geophys. Un. 21,287 (1940). 21. S. K. RUNCORN,Encyclopaedia of Physics XLVII, 498 (1956). 22. T. NAGATAand T. R~TAICE, J. Geomag. GeoeZect. XIV, (4) 213 (1963). 23. G. GOERTZEL,Amer. Math. Monthly 65,34 (1958). 24. R. J. HADERand A. H. E. GRANDAGE,Symposium on Design ofIndustrial Experiments (ed. E. V. Chew), p. 108, Institute of Statistics, Raleigh, N. Carolina (1956). 25. L. H. C. Tmpm-r, The Methods of Statistics, p. 387. Wiley,New York (1952). 26. E. WHITTAKERand G. ROBINSON,The Calculus of Observations, p. 282, Blackie(1944). 27. R. B. BLACKMANand J. W. TUREY, The Measurement of Power Spectra. Dover (1958). 28. R. Bock and W. S-ANN, Katalog der Jahresmittelder magnet&hen Elemente der Observatorien tmd der Stationen, an denen eine Zeitlang erdmagnetischeBeobachtungenstattfanden. Berlin (1948). 29. T. NAGATAand M. SAWADA,J. Geomag. Geoelect. XV, 2 (1963).
h'dIOM~&IaJIKB AaHHbIX reOMarHHTHhIX BeKOBbIX KaMeHeHHii B TpeX EUlebfeHTaX X,Y K Z B IIFITH06CepBaTOpHJIX paaMeIgeRKbIx II0 SeMHOMy ILlapy,IlOKaabIBaeT,9TO aMIlJIl¶TTy~L.I,lIOJIj’%HHW3 B 6O-TKJIeTHEi~gliKJI~OCpe~CTBOMrapMOHEl~eCKOrO aHaJIEl8a, BeCbMaLUEFlHTeJIbRM. kIJI0 06KapyHCeH0,9TO aMIUIHTJ'AblI'apMOHtlK6onee BbICOKOrO 3TOT gHKJI MOX-330 mui Boo6Ige HeaHaUUTeJIbm: nopsnKa McKee aKawTenbHhl, HeIIOCpe~CTBeHHOBH~eTbH'drpa@l¶KOB~aHHbIXHa6~IO~eHH~. 3THpeayJIbTaTbIUOATBepB hoxAaroTcn aKamfaou cmoBor0 cneKTpa. ~OJIb8yflCb J'PJIaMH@aa, IlOJryreHHbIMki THJIeTHHt (lIOJIJ'CTOJIeTHHt)gKKJInO AaHHbIM pa8JIEiWMXlIepEiO~OB,6oJIee J-TOgHeHHbIe aKaverim~n~aTorogKKnaoKaaanmbpaBHmo~KMnCs 61 f 6 JleT. ~pH6JUWiTeJIbHbIe IlOJlyCTOJleTHUeBHWIeHUR ElHTepeCHbIB BKAy TeOpeTHreCKOti BOBMO)KHOCTH Kone6anmi nona SaOnepHoAa B 77 neT,HanaraeMoro Ka rnamoe none.
Space Sci. 1965.
Vol. 13.
pp.1097
to 1110.
Pergamon
Press Ltd.
Printed
in Northern
Ireland
SOME MORPHOLOGKAL ASPECTS OF GEOMAGNETIC SECULAR VARIATION Department
L. SLAUCITAJS* and 33. E. WINCH of Applied Mathematics, Unive~i~ of Sydney, Sydney, N.S.W., Australia (Received 9 June 1965)
Abstract-Analysis of geomagnetic secular variation data, in the t&e elements X, Y and Z at five observatories distributed over the Earth, shows that amplitudes obtained for a 50 year cycle by harmonic analysis are highly significant. Amplitudes of higher order harmonics are found to be either not as significant or not significant, indeed, the cycle can be seen dire&y on the graphs of ob~ational data. These results are confirmed by power spectrum analysis. Using phase angles obtained for the 50 year (semicentennial) cycle from different stretches of data, an improved value for the period is found to be 61 f 6 years. The approximate semicentemkl values are interesting in view of the theoretical possibility of an oscillation of an S,O field of period 77 years superimposed on the main field. INTRODUCTION
of the author@ indicated a “pulse” or “period” of appro~mately 50 years in secular variation of the geomagnetic field. This was confirmed by independent papers of Barta@“) and more recently by Gaibar-PuertaC-‘1 and Slaucitajs.(8) Wasserfall,(*) using data for Oslo 1820 to 1948 mentions some “nutational” period of 70 years, Slaucitajs(l”) made a harmonic analysis of 44 years of secular variation data from a South American station, so that a fourth harmonic represented the effect of the solar sunspot cycle. Stoyko@l) indicates that the fluctuation in terrestrial time in a period of about 50 years is correlated with secular change in the total intensity, and Elsasser(12) comments that periods of 40 years are fairly representative of the secular variation. For the IAGA Meeting in Toronto, 1957, a communication on forecasting of secular variation was presented by Kalinin(13) in which the author mentions that part of the changes in the geomagnetic field effected by internal causes has a quasi~riodical character at observatories in Eurasia with a period of a few decades and that another part has an eleven year period connected with the geomagnetic activity. Yukutake(14) indicates that analysis for short intervals of secular variation data reveals a 50-year and 25-year period, Depietri(15) analysing secular variation for the Northern Hemisphere between 350” and 80”E received a period of 55 years. Recently Degaonkar(16) analysed AH, AZ and AF for three American stations with respect to the 11 year sunspot cycle, and Yukutake’s paperQ7) on the eleven year period variation in the geomagnetic field associated with the solar activity was presented to the IAGA Symposium in Pittsburgh. These researches are largely empirical, and one must be cautious since(l@ periodicities found by harmonic analysis and not predicted by previous theoretical considerations may be untrustworthy, as many complications are capable of giving spurious periods. As we know Bartels’19) indicated the predominantly local character of secular variation and after McNish(20) it is preferred to relate the major part of secular variation with the residual, non-dipole field. Nevertheless the morphology of the phenomenon shows that some periodicities in secular variation exist(21)and one of these is the approximate semicentennial. One
* On leave from the National University of La Plats, Argentina. 1097
L, SLAUCTFAJS and D. E. WINCH
1098
The physical theories of the origin of the main field and secular variation field based on magnetohydrodynamics need to meet the morphological aspects shown by and deduced from observational data. Now, Nagata and Rikitake(22) indicate the theoretical possibility of an oscillation of an S,Ofield about the main field with a period of 77 years. Although one could not conclude that the observed cyclic changes in AX, AY and AZ (see below) are directly TABLE 1
observatory
Geographic Long. Lat.
1. Wyssokaja Doubrawa
56” 44’N
61” 4’E
2. Niemegk
52” 4’N
12’ 41’E
3. Pilar
31” 40’S
296” 7%
4. Cheltenham
38” 44’N
283” 1O’E
5. Kakioka
36” 14’N
140” 11’E
FIG. 1. GEOMAGNETIC COORDINATESOF OBSERVATORIES WHOSE SECULAR VARIATION DATA ARE U~~~A~AL~S.
related to this quadrupole oscillation suggested by the theory, it is most interesting to find secular changes with a period comparable to the theoretically suggested one. It is the purpose of this paper to present harmonic and power spectrum analyses of the armual mean rate of change of the elements X, Y and 2, denoted AX, AY and AZ using all available data at the magnetic observatories listed in Table 1. Figure 1 illustrates the distribution of these observatories over the Earth. In the well-known graph of Declination and Inclination for London from the 16th century, now expanded till mid-20th century by Gaibar-Fuertas(6) in which short period features are not obliterated to give prominence to an apparent 500 year cycle, a superimposed cycle of se~centenni~ period is also suggested.
SOME MORP~OL~IC~
ASPECTS OF GEO~GNE~C COMPUTATIONAL
SECULAR V~A~ON
1099
DETAILS
First the graphs of observed values of AX, AY and AZ and of running eleven year means to eliminate the solar activity period were prepared. These graphs suggested (as already reported by Slaucitajs(l)) testing the period mathematically. The secular variation corresponds to first differences of the main field values, and so does not contain any linear trend in the latter, so that any superimposed oscillations are more readily seen. For intervals of data longer than about 70 years, any parabolic and cubic trends in the main field result in linear and parabolic trends respectively in the secular variation field which distort any oscillations. For this reason, harmonic analysis of xxx&r variation values over 100 years to test the hypothesis of a 50 year cycle will not lead to significant results. An attempt to fit by the Method of Least Squares the regression model 4
Y = a0 +kslak sin mkt + $ bKcos mkt k=l
where m = 27~150to given dataye, at equally spaced intervals t = 0, 1,2, . . , , N to the normal equations
(1) 1 leads
If now N is taken to be 50,100,150, . . . then these normal equations reduce to diagonal form, and estimates of a, and bBwill be entirely uncorrelated; from a statistical viewpoint this is very desirable. In each analysis N has been taken as 50. The solution of the normal equations is now
2iv-1
a, = -
yr sin mkt NC t-0
b, = ;;i;yt
cos mkt
k = 1,2,3,4
k = 1,2,3,4.
A computing algorithm given by Goertzel WI) takes one tenth of the computer time required for direct evaluation of a, and bk using sine and cosine subroutines and this has been as follows: with
ET,=0
1100
L. SLAUCITAJS
and D. E. WINCH
compute for k=N-
1, N-2
N
-z bk = yO f N 3ak
=
,...,
I,
V, cos mk - U, U, sinmk
Since the sum of squares due to regression is advaO + 2 adad2)
+ 2 bk(bkN/2)
hence the residual sum of squares is
where .r2 is an estimate of the variance around the regression relationship. Confidence limits for a, are a,, f tg l/N and for a, and bKare ak f 6 and b, f 6 where 8 = t&2/N),(“) where t, is found in Student’s t table with N - 9 degrees of freedom.(26) The expression (1) may be put in the form
a, + i Cksintmkt + #kl k=l
Noting that annual mean values given for the years x and x + 1 have a first difference corresponding to the secular variation at epoch x + 05, it proved convenient to refer all phase angles to epoch 1900.5. From confidence limits a, rt: 6 and bk f 6 for a, and 4, confidence limits for C, and &.k,confidence limits for C, and & are given by Whittaker and RobinsorP) as C, f 8 and 46, f s/C,. The quantity s/c& will be in radian measure, The results of this analysis are given in Table 2. For those cases in which the 50 year cycle is s~g~fi~nt and higher harmonics not as sibilant, or insig~ficant, the following theory to indicate a more suitable value for the period of the cyole is applicable. The problem is to fit the curve y=Asinkt+Bcoskt = C sin (kt + a) where k = 24T and A + iB= Cexpia
(2)
SOME MORPHOLOGICAL
ASPECTS OF GEOMAGNETIC
SECULAR VARIATION
1101
to data which is really of the form y = D sin (k’t + p) where k’ = 277/T’
111 0
E g
20
WYSSOKAJA
10 1
AX
DOUBRAWA
I
L
-50
s. 10 c. 0E 0 ‘D > 20
20 10 0 -10 E -20
, 1990.5
I
v
I
1900.5
1910.5
I
I
1920.5 Epoch,
1930.5
1
I
1940.5
19505
I
19605
yr
FIG. 2. WYSSOKAJA DOUBRAWA. UNRROKENLMEISSECULARVARIATION,DASHED LINRISREOREWON CURv@FORFIRSTFWTYYEARS,DO~DLINEISREGRESSIONCURvEFORLASTFIFNYEARS.
Solving the least squares equations for A and B, and substituting into (2) gives C exp ia = s [ (exp i[p + (N -
-
l)(k -
{exp -U
k’)/2]} zz Nii:
+ (N -
1 $
l)(k + k’)P])
if
Nti
I::$]
Assuming k w k’, the first term of the expression dominates, and hence a M B + (N - l)(k’ - k)/2
(3)
If a second analysis is performed on a further set of observations, whose tist observation is the mth after the fist observation of the fist set, and a, is obtained as the phase angle referred to the point t = 0, then a2 + km = /? + k’m + (N -
l)(k’ - k)/2
(4)
L. SLAUCITAJS
1102
and D. E. WINCH
Solving (3) and (4) for k’, the expression for T’ is obtained: T’ = 360mT/{(cc, - a)T + 36Om))
(5)
where as and a are in degrees of arc, and T is 50. If confidence limits for a and a2 are a & 6, and a2 f 6, respectively, then confidence limits for T’ are given approximately by T’ f. d(d,Z + 6,2)T’2/(360m) NIEMEGK
70
60 50 g
40
:
30
sl :
.
20 10
;; b 2.
50 -
i
403
0 m
AZ f
30 zo-
\
lo-
, \
O-10 -20
-
-30 -40
-
-50
-
-60
-
-70 -90
-
-90 -
f 1990-s
1
1900 5
I
1910.5
I
Epoch,
FIG.
3.
I
1930.5
1920.5
I
1940.5
t
1950.5
I
1960.5
yr
NIEMEOK. UNBR~KENLINEISS@CULARVARUTION,DASHED LINEIS REGRESSIONCURVE FORFIRSTFlFTYYEARS,DO-fTEDLINEISRRGRESSIONCURVEFORLASTmFIYYEARS.
The results of this analysis are included in Table 2, except for observatory 3, Pilar, where m was small and the results were non-significant. Although 95 per cent cotidence limits are given for amplitudes and phase angles, confidence limits for the improved period are approximately 60 per cent, due to the nature of the approximation made in the theory. Twenty-five autocorrelation coefficients were computed for the power spectrum analysis, since this is the smallest number which would give a spectral peak corresponding to a
SOME MORPHOLOGICAL
ASPECTS OF GEO~AG~E~C
SECULAR
VARIATION
fifty year cycle. The cosine transformation(a7) of the autocorrelation computed using the Goertzel algorithm described above. SOURCES
1103
coefficients was
OF DATA
The principal sources were the Bock-Schumann catalogue@‘) and the Nagata-Sawada catalogue,@) Annual mean values in the interval not covered by these two catalogues,
6
-60
E -70 E -80 ::
5 Ti -50 tj - -60 > -70 s 2 x
*
-so 40 30 20
.
10 0 -10 -20 1910*5
t920*5
19305
1940 5
1950~5
19606
Epoch. yr
Fm. 4. PILAR. UNBROK.ENLBIBI~SECULARVARIATM)N,DASHEDLINEISREGRFS~IONCUR~EFOR FIRST FWIT YEARS,DOTTED LINE IS REGRESSION CURVE FOR LAST FIFTY YEARS.
1949 to 1955, were obtained from yearbooks of the observatory concerned, or from the institute controlling the observatory. Mr. B. R. Leaton very kindly supplied missing values for Wyssokaja Doubrawa. CONTUSIONS
Power spectrums and harmonic analyses indicate that the amplitude of a 50 year cycle is highly sign&ant at the five observatories distributed over the Earth, except for AY and AZ at Kakioka; in the case. of AZ this seems to be due to the great variability of the earlier values. The amplitudes of the higher order harmonics are either not as significant or not significant. The average result for the improved values of the periods is found to be 61 ,j, 6 years(60 per cent confidence limits). This result taken with the theoretical prediction of an oscillation of an axial quadrupole field having a period of 77 years, suggests that spherical harmonic coefficients of the main field should be computed at at least two year intervals, using all the data contained in the
106f 3 218ir 14 156i: 9 12Of7 80 f 17
255rfr:8
15 k 1
26 & 6
27 i 4
20 i 2 11 f3
21 + 1 16 &2
20 f4
40 l 3
O&2 -312
44 f 1
47 a 2
AY 18905-1939~5
1912.5-1961.5
14 f 4 12 f 4
16OrtlO 336f 15 324& 16
20 f 3
16 f4
16 + 4
312
-50 xt 3
-52&3
1908%1957.5
3. Ax 1905.5-19545
8zt9 10 f 3
221f 17
29 + 9
8k6
19125-1961.5
411 6f2
10 f 2 63~3
6f4
13~6
2&l
l&l
6&3 13 + 5
G P
AZ:1890*5-1939.5
19125-1961~5
2. AX 1890+-1939.5
1911*5-1960.5
AZ;1887.5-1936.5
1911+-1960-5
279f 3
105f 3
21 & 1
2&l
-3 + 1
AY 1887S1936~5
142i 5 106$: 10
lPll+-19605
1. AX 18875-1936.5
29 It 5
41 deg.
32 & 3
c, li
-321lz4
a0 li
lztl 3f2 6rtr9 5&3
301+ 14 148f 22 308f 63 343i 20
295+ 21
514
5zt4
4xt3
16 f 32
306f 17
3rtr2
143i 14
2&6 7f4
70 i 40
201f 55
240f 54
338f 44
88 & 75
101+ 37
274f 61
326+ 45
222f 46
350i 34
115f 217
120rt 22
298i. 17
3+1 2331
359i 65 318i 31
4s deg.
2f3 IO & 5
G P
31 f 375
132It 26
2134 44
11 f23
147i 27
#e deg.
2414 44
7f5
5+6
5*4 354
2rt3
4It9
3f2
3&l
413
3&2
4&4
234& 51 201i_ 93
195f 113
138f 117
58 f 41
125i: 19
166& 50
225f 41
109i-66
179f 67
109zt86 134f 17
288& 51
3f3
l&l 3ztl
164 deg.
81 f 16
59 zt 4
Improved period, yr
~ISPIL~,OBSERVATORY
G ?
OBSERVATORY~~SWYSS~KAJADOUBRAWA,ORSERVATORY~ISNIEMEGK,OBSERVATORY 4IS ~L~~M,O~~VA~RYsIS~OKA
-38f2
Ob. El. hterval
INDICATED ~ERTHEHIWXN~‘?NTERVAL”.
TABLE2. NINETY-FIVE PERcENTCONmDENCEL~sFORAMPLITUDESINGAMMASPERYEARANDPHASEANGLESINDEGREESGIVENATEPOCH 1900.5 OPFOUR HARMONIcSOFA%YEARFUNDAMENTAL. %XTYPERCENTCONFIDENCELMTSFORTHElMPROVEDPERIOD. THEFIFTYYEARINTERVALSIJSEDFORANALYSISARE
? $ h
r
-57 ri: 1
1908s1957.5
-20 f 5
-9 f 1 -9 i 1
1911.5-1960.5
AY 1902.5-1951.5
-17$:2
AY 1897.5-1946.5
19W5-1958.5
AZ 18975-19465
4 j, 15 -2 rir13
-16 & 1
7&4
1909.5-1958.5
6&6
1909.5-1958.5
-53 xk5
1911.5-19605
5. AX 1897.5-1946.5
-56 rt 5
AZ 19025-19515
1911*5-1960.5
-37 ic 4
4. AX 1902.5-1951.5
AZ 1.905*5-1954.5 -12zk2 19065-1955~5 -12 * 2
-58 zk1
hY 19055-1954.5
22 -+ 18
18 i22
l&2
4f2
15 f 6
13 zk8
46 f 7
44 + 7
10 i 2
10 It 2
53 f 7
40 f 5
24 c?c 3 23 f 3
13 12
13 & 1 l&l
171j, 47
142& 70
356& 169
240It 35
115* 21
119& 36
120-+:8
125+9
171& 20
174& 9
92 f 7
129rt 8
5 i 22 6 It 18
4rrt2
i&2
5&6
3f8
41t7
517
4rt2
5*2
19 f 7
24 f 5
3&3 2xt3
346f 7
2&2
349Jr 7
1391:7
143$5
18 f 234 144i 172
24 z!c29
109z!x194
112f 68
143f 160
145&- 82 96 &9l
227f 23
218& 20
135f 12 59 & 21
343& 93
357zt 59
350f 55
355f 110
3 i 22 8 -4r18
3f2
2rf2
2f6
J&t?
7&7 10 & 7
3f2
4f2
4rt;5 17 St 7
6rf3
6+3
2-rt2
21-l
79 Ii=31
l&l
167rir57
326f 376 100+ 133
10 + 46
127f 66
244 & 174
241 f 93
358f 40
2 i 61
93 f 35
96 i 26
318rt 23
88 f 83
318zk66 318rt 22 212It 64 299* 102
19 $I 22 10 & 18
31 rfr:77
4f6 2&2 5&2
333xk491
lct8
2614 64
244f 77
5&7 6f7
139f 58
114f 52
212 2f2
234I: 31
5kt5 13 rfl:7
167&56
239f 34 225I 36
5f3 5k3
201+ 52
229f 26
212
240+ 29
55 f 59
37 f 27
21 & 18
53 ct 27
55 f 11
53 zk 12
117f 45
1106
L. SLAUCITAJS and D. E. WINCH
50
CHELTENHAM AX
30 20 10 40 0E -10 ~ \ -20 \ \
-30 -40 -50 -60
[
; -70 E -80 ~ 8l
.
10 i 0 ;; - -10 & > -20
AY 1
-90 -100 -110 -120 -130I__-L_i_-_. 1900.5 1910.5
1920.5
1930-S
1
I
I
1940.5
1950.5
1960.5
Epoch, yr FIG.~.~IELTDIHAM.
UNBROKEN LINE IS SECULARVARIATION,DASHEDLINEIS REGRESSIONCURVEFOR~TmFTYYEARS,DO~D~ISREORESSIONCURVB FOR LAST FIFTYYEARS.
catalogues mentioned above, and examined for superimposed periodicities. Least Squares applied to a model of the form
y =
2 a*xf
i-o
The Method of
+ anfl cos (27rxfT) 4 aS.+zsin (2427
where ordinates y, are the spherical harmonic coe5cients of the same kind, x is the year, and T is the period, would be appropriate. From the paper by Nagata and Rikitake (s2)the period of oscillation of the SBofield is deduced as 23ra 1 75-p T=zg J( 3 )
SOME MORPHOLOGICAL
I
_---__L*;--5 r’ 0” 4r
P
.:
1 . . . . . L ,.................:
l_______,____ix.I~z
SECULAR VARIATION
,____.i
:r-_--i...
i I : I :
ASPECTS OF GEOMAGNETIC
:..... .. .
I
,.......,.....................
-...
?
ci
i
I
’
I
ii, “0
I
I
“0 Ad~/~(sourwo6)~X~~suep
0
JaMOd
I
5:
1107
-
LA
__
: :.
Period.
56
,:
I
2.9
I.,
Ii
II,31::fL
Ll
i.f
:.!
. . . :: i
.
. . .
i
yr
:
1.: . ..:
:
’
2.0
I
I
I
;
I
L ,
,I;;
i
: : : :
:” . . . : : :
I
Ir’ I:
:: :: :: :.;
i”
1
I.
I
:.
FIG. 8. NIEMEGK. UPPER 90% CONFIDENCE ~rmrf? ARE SHOWN FOR EACH ELEMENT.
A12.5i
A-: : : : : : . . : : : : : : : : . : : : : : : :.:
: :
I:
-1
I....
Fwiod.
yr
’
(
i
LIMITSARESHOWNFOREACHELEMENT.
FIG.9. PILAR. UPPER 90% CON~DENCE
_r?
AV .............A2
AV ,...,........AZ
3 Ax
Spectrum ________
Spectrum 2 ______ -_ A)(
5 m
c
SOME MORPHOLOGICAL
ASPECTS OF GEOMAGNETIC
SECULAR VARIATION
yz
. ..... ..... ............ ... r______-_-____ ___ ..:
....:
,__-__J
:......
X>N
3
: .. . .. . .
-ada
L____,
:*.: .. . . . .. ,. .
. .. .... . :..
.
L-_______,
r_-___, r_____J
: . . . . .. ..
I :
L_
I
.i i
L____?
,-_______J
r
rd
: . . . . . . . . ra . . . .
,
:......... ...:
I
‘6
-6
Ida /z(
SDWWD~ )‘Xa!suap
P Jarnod
1109
1110
L. SLAUCITAJS
and D. E. WINCH
where a is the radius of the core, p the density of the core, S, a steady poloidal field in the core. When T = 50years, the value for S, is found to be 2.8 e.m.u., a not unreasonable value. REFERENCES 1. L. S~~vcrr~a, Contri6. Baltic Univ. 63, 1 (1948). 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
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