Centered and Eccentric Geomagnetic Dipoles and Their Poles,

Transcription

1 REVIEWS OF GEOPHYSICS, VOL. 25, NO. 1, PAGES 1-16, FEBRUARY 1987 Centered and Eccentric Geomagnetic Dipoles and Their Poles, A. C. FRASER-SMITH Space, Telecommunications, and Radioscience Laboratory, Stanford University, California Using a unified approach, expressions are derived for the various pole positions and other dipole parameters for the centered and eccentric dipole models of the earth's magnetic field. The pole positions and other Parameters are then calculated using the 1945-i985 International Geomagnetic Reference Field Gauss coefficients and coefficients from models of the earth's field for earlier epochs. Comparison is made between (1) the recent pole positions and those pertaining since 1600 and (2) the various theoretical pole positions and the observed dip pole positions. 1. INTRODUCTION In response to questions concerning the most recent positions of the various magnetic poles in the Arctic and Antarctic, where the Space, Telecommunications, and Radioscience Laboratory is currently operating ELF/VLF noise measurement equipment [Fraser-Smith and Helliwell, 1985], I recently computed the positions of the poles for the centered dipole and eccentric dipole models of the earth's magnetic field, using the Gauss coefficients for the nine available main field models of the International Geomagnetic Reference Field (IGRF) for the years i , as tabulated by Barker et al. [19863 and Barraclough [1985-]. In the process of deriving the pole positions I also computed, for each IGRF field model, the scalar moment and orientation of the centered and eccentric dipoles and the; position of the eccentric dipole. Further, in order to provide some perspective on the likely changes in pole positions and other geomagnetic dipole parameters over the next few decades I extended the computations to representative earlier years for which the necessary Gauss coefficients were available' the results of these computations, when combined with those for the IGRF data, give a comprehensive picture of the changes in pole positions and other dipole parameters that are likely in the near future. Since this updated information on pole positions and other properties of the centered and eccentric dipoles does not appear to be readily available and is of general interest, I present it here, along with some details of the computations. The eccentric dipole computations are based on formulas originally derived by Schmidt!-1934-] and described in English by Bartels ] and Chapman and Bartels [- 940]. They were used by Parkinson and Cleary [1958], whose derivation of the details of the eccentric dipole for epoch 1955 provided a model for this work, and they require only the first eight Gauss coefficients in each spherical harmonic field model. To illustrate, Table 1 lists the first eight Gauss coefficients for the 1980 and 1985 IGRF models; a full list of the coefficients through 1'0 orders (rn = n = 10) is given by Barl er et al. [1986] and Barraclough [1985]. In accordance with modern practice the coefficients listed in Table 1 are given in nanoteslas, and I similarly use SI units throughout the derivation of dipole parameters, which necessitatesome small changes in the original formulas [Schmidt, 1934; Bartels, 1936' Chapman and Bartels, 1940]. It will be assumed throughout this work Copyright 1987 by the American Geophysical Union. Paper number 6R /87/006R that the Gauss coefficients are in Schmidt-normalized form (unlike the Gaussian normalization used by Jensen and Cain [1962], for example). Because the IGRF coefficients are predominantly given to four significant figures, and the larges is given to five figures, I will usually provide four significant figures for the dipole parameters computed from the IGRF field models, and I will assume that the fourth figure is meaningful. The International Association for Geomagnetism and Aeronomy (IAGA) is primarily responsible for the IGRF, and it has been particularly active in recent years with revisions of past field models, the issuance of new IGRF models for past and current years, and the provision of Definitive Geomagnetic Reference Field (DGRF) models, the l,tter consisting essentially of IGRF models that have been revised and probably will not be altered substantially in the future. The first IGRF model was adopted by IAGA in 1968 for the main field at epoch [Peddie, 1982] and the current, or "fourth generation," IGRF now includes IGRF models for 1945, 1950, 1955, and 1960; DGRF models for 1965, 1970, 1975, and 1980; and an IGRF model for 1985 [Working Group 1, 1981; Barraclough, 1985; Barker et al., 1986]. The data in Table 1 are taken from the DGRF 1980 and IGRF 1985 models. IAGA's activity is undoubtedly having a strong influence on studies of the earth's magnetic field, and the fact that updated reference fields are likely to be issued more regularly in the future than has been the case in the past has influenced this work. The traditional approach in papers treating the centered and eccentric dipole models of the geomagnetic field is to list computed properties of the dipoles without specification of the mathematical procedures that are involved in their derivation. This approach saves space but makes it difficult for researchers interested in computing up-to-date values of magnetic fields on and above the earth's surface according to either of the dipole models to do so without extensive reference to the literature. It is, in fact, very simple to obtain the centered dipole parameters from the spherical harmonic representations, but the procedures are no longer well documented and can be time consuming to retrieve. The eccentric dipole parameters are more difficult to compute, and the procedures appear never to have been completely documented. Further, one of the best descriptions of the eccentric dipole approach to modeling the earth's magnetic field contains an error (see section 3.1). I have therefore described the steps required to obtain the dipole parameters from the Gauss coefficients that they may be quickly computed from future IGRF or DGRF field models. Another problem faced by a nonspecialist desiring to utilize

2 2 FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES TABLE 1. The First Eight Gauss Coefficients in the 1980 and 1985 Field Models of the IGRF n m gm n hm n gm n hm n , , ,956 5, ,903 5, , , , ,129 3, , , , The units are nanoteslas. the dipole models to obtain up-to-date values of the earth's magnetic field involves the coordinate systems in which they are defined. It is not easy to obtain the magnetic field at a given geographical position (which is likely to be the most common requirement) from a simple listing of the dipole parameters. Changes of coordinate systems are required (one change, a rotation, for the centered dipole; two changes, a rotation and a translation, for the eccentric dipole) that can be time consuming and difficult for someone not freshly acquainted with the procedures involved. In this work, in addition to listing the dipole parameters and showing the current pole positions, I document most of the steps required to obtain the magnetic field components at any geographical location from either the centered or eccentric dipole models of the earth's field. Finally, it is well known that the earth's magnetic field is undergoing a secular variation [e.g., Parkinson, 1982; Merrill and McElhinny, 1983], and it is of course because of this variation that updated dipole field parameters are required from time to time. The change can impact significantly upon the choice of locations for certain measurements within a decade or two [e.g., Stassinopoulos et al., 1984], which is well within the professional lifetime of a scientist. Thus in addition to providing up-to-date dipole parameters I have also endeavored to put the parameters into an historical perspective by briefly indicating some of the changes that have taken place in the parameters over the last few centuries. Much has been written on these changes [e.g., Adam et al., 1970; Barraclough, 1974; Dawson and Newitt, 1982] and on the changes that have taken place over larger time scales [e.g., McElhinny and Senanayake, 1982]; my purpose is to indicate the direction of the changes that are likely over the next few decades Coordinate Systems 2. CENTERED DIPOLE Two basic coordinate systems are used in this work. The primary, or reference, system is based on the earth's geographic coordinates. Some variation of choice is possible; I will assume that it is a geographically based spherical polar coordinate system with its origin at the center of the earth (assumed spherical), in which the position of a point P is given by (r, 0, p), where r is the radial coordinate, 0 is the polar angle measured from the north polar axis, and b is the azirduthal angle, equivalent to the longitude, measured to the east from the Greenwich meridian (Figure 1). Thus 180 ø > 0 > 0 ø, and 360 ø _> b > 0 ø. The angle 0 is the colatitude and is related to the geographic latitude 2 through 0 = 90 ø- 2. It is usual for southern latitudes and western longitudes to be given a negative sign, and that convention will also be used here. The geographic rectangular coordinate system x, y, z, also shown in Figure 1, will not be used widely in this work, since spherical polar coordinates provide a simpler representation when spherical geometries are involved. However, the rectangular system is the conventional reference for the position of the eccentric dipole. From the discussion above it can be seen that the positive x axis points toward 0 ø of longitude, the y axis points to 90 ø east longitude, and the z axis points to the north. The other basic coordinate system is a spherical polar coordinate system based on the centered magnetic dipole. In this system the field is symmetric about the axis of the dipole, which, as indicated by the description, is located at the center of the earth, and the position of a point P is given by (r, O, ), where r is the same radial coordinate as in the geographic system, O is the colatitude measured from the centered dipole axis in its extension through the northern hemisphere of the earth (the centered dipole latitude, denoted by A, is given by 90 ø -O), and ß is the longitude measured eastward from the meridian half plane bounded by the dipole axis and containing the south geographic pole. A variety of coordinate systems are used in the literature to describe the geographic and centered dipole systems, so it is important to note the conventions involved here: The basic coordinate systems are both spherical polar, and with the exception of the common radial coordinate r the geographical coordinates are denoted by lowercase symbols and the centered dipole coordinates by the same symbols in uppercase. It is common for the coordinate pair (O, ) or equivalently (A, ) to be referred to as the "geomagneti coordinates" of a point on the earth's surface [Schmidt, 1918, 1934; Chapman and Bartels, 1940; Matsushita and Campbell, 1967; Parkinson, 1982] and for the two points where the axis of the centered dipole crosses the surface of the earth to be called the "geomagnetic poles." This restriction of the general term "geomagnetic" (that is, denoting "relative to the magnetism of the CD Axis x B ' z I I N, y Fig. 1. The geographically based spherical polar coordinate system r, 0, b that is used as a reference in this work for the CD coordinate system. In the associated Cartesian system x, y, z the positive x axis points to 0 ø of longitude, the positive y axis points to 90 ø east longitude, and the z axis points to the north. The coincident origins for the two systems are located at the center of the earth, O. Only the northern part of the CD axis is shown; its intersection B with the earth's surface, represented by the sphere r R e in this figure, is the north CD pole (R E, O n, Pn)' N is the north geographic pole, and P is a general point.

3 FRASER-SMITH' CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES 3 earth") to the special case of the centered dipole model of the earth's field has disadvantages, as pointed out by Chapman [1963], and in this work the problems pointed out by Chapman are even more acute because of the use of two different dipole models for the earth's field. I therefore build on Chapman's suggestion (also see Matsushita and Campbell [1967]) and, instead of "geomagnetic," use "centered dipole" (or CD) and "eccentric dipole" (or ED) to describe quantities relating to their respective field models. Thus the CD poles are the intersections of the CD axis with the earth's surface, with the north CD pole being the intersection in the northern hemisphere. The geographic and CD coordinates can be related through the use of the cosine and sine rules for spherical trigonometry, as is shown by Chapman and Bartels [1940] and Mead [1970], in particular. To effect a transfer between the coordinate systems, it is necessary for the orientation of the magnetic axis of the centered dipole to be specified in the geographic coordinate system. I will denote the orientation of that part of the magnetic axis intersecting the earth's surface in the northern hemisphere by 0,, 4, (Figure 1) and of the part intersecting the surface in the southern hemisphere by 0 s, (ks, where 0 s = 180 ø -0, and 4 s = 180ø + 4. The distinction may seem trivial, but it is a primary source of confusion in computations of the earth's magnetic field from the dipole models because of the confounding circumstance that the southward directed pole of the dipole is actually a north magnetic pole and the part of the magnetic axis extending out from the north pole of the dipole actually intersects the earth's surface in the southern hemisphere. It follows from the above choice of notation that the coordinates of the north CD pole are (R e, 0,,, b,,), and for the south CD pole they are (R e, 0 s, Cks). A useful quantity in CD field computations is the CD declination ½. It is an idealization of the conventional declination used in geomagnetism, which is defined to be the angle between true north and magnetic north, taken to be positive when magnetic north is to the east of true north. In CD case it is the (spherical) angle between geographic north and the north CD pole, taken to be positive when the CD pole is to the east of geographic north. Applying the sine rule to the spherical triangle on the earth's surface defined by the point P, the north geographic pole, and the north CD pole (Figure 2), we obtain sin 0 sin 0, sin (9 - - (1) sin (180 ø -- ) sin (--½) sin (& -- &.) In addition, the cosine rule gives cos 0,, = cos 0 cos 0 + sin 0 sin 0 cos (-½) cos 0 = cos O. cos 0 + sin O. sin 0 cos (& - &.) (2) cos 0 = cos 0,, cos 0 + sin O. sin 0 cos (180 ø - ) From these equations we obtain 0 = cos- [cos O. cos 0 + sin O. sin 0 cos (& - &.)-[ ß = cos- [-(cos 0 - cos O. cos O)/sin O. sin O] ti) - sin- [sin 0 sin (4 - bn)/sin (9] (3) ½ = cos-x[(cos 0, - cos 0 cos O)/sin 0 sin (9] ½ = sin-x I-sin 0, sin (4-4,)/sin (9] Given any point P with geographic coordinates (r, 0, 40, the B, Fig. 2. The spherical triangle used to convert between geographic and CD spherical polar coordinates. N represents the north geographic pole, B is the north CD pole, and P is a general point with geographi coordinates r (= Re), 0, b. above equations provide its CD coordinates (r, (9, ti)) and the CD declination ½. Two different equations are given for each of ti) and ½ in order to avoid the ambiguity in angle that occurs when an inverse sine or cosine is evaluated for a possible angular range of -180 ø to + 180ø: For each value of the argument there are two possible angles (for example, cos can be either 20 ø or -20ø). The ambiguity is unimportant if a guide to the expected values is available (a world map of CD coordinates, for example). However, if the computations are being conducted without such a guide, both the inverse sine and inverse cosine should be computed, giving two pairs of possible angles; the correct value is the one angle that is common to the two pairs. The ambiguity does not occur for (9 in (3) because its range is restricted to 0ø-180 ø. The inverse transformation, from CD coordinates to geographic, also follows from (1) and (2); the relevant equations are 0 -- COS- I[COS O n cos (.4- sin O n sin 0 cos (180 ø - 4 = 4 n 4- cos- [(cos (9-- cos O n cos 0)/sin O n sin 0] (4) 4 = 4 + sin- [sin (9 sin ti)/sin 0] 2.2. Derivation of Centered Dipole Parameters The parameters of the centered dipole model of the earth's magnetic fields are specified completely by the first three Gauss coefficients gxo, gxx, hxx. The formulas required for the derivation of the moment M and orientation 0, 4 n of the north magnetic axis of the centered dipole in the geographically based spherical polar coordinate system are Bo 2 = (g o)2 + (g )2 + (h )2 cos 0. = --g ø/b o (5) tan ok. = h /g where B o, a reference magnetic field (termed the "reduced moment" by Bartels [1936], in a different system of units), gives the dipole moment M through the equation 4 M = BoRe 3 (6) /.to where R e is the radius of the earth, which will be assumed to

4 4 FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLEG have a mean value of km, as specified by Geodetic Reference System 1980 [International Union of Geodesy and Geophysics, 1980]. Substituting the DGRF 1980 values of g2 ø, g22, and h22 from Table 1, (5) gives B o x 104 nt M x 1022 A m 2 0 n= ø Pn ø Similarly, substituting the IGRF 1985 values of g2 ø, g2, and h22 from Table 1, (5) gives Bo x 104 nt M x 1022A m 2 o n ø Pn ø Given the above 1980 values of 0 nand Pn the following geographic coordinates are obtained for the 1980 centered dipole (geomagnetic) poles: north CD pole is 78.81øN, 70.76øW, and south CD pole is 78.81øS, 109.2øE. Similarly, from the 1985 values of O n and Pn the geographic coordinates obtained for the 1985 centered dipole pole are north CD pole of 78.98øN, 70.90øW, south CD pole of 78.98øS, 109.1øE Centered Dipole Magnetic Field The magnetic field components produced by the earth's equivalent magnetic dipole take their simplest form in the CD coordinate system, since the field is symmetric about the axis and there is no dependence on azimuthal angle. In terms of the CD (or "geomagnetic") coordinates the centered dipole approximation to the earth's magnetic field takes the form om(3 cos ) 2/2 IBel - 4 rr 3 2 om cos O (7) (8) (Be),. = -- 4rrr 3 (9) (Be) o = -- o M sin {9 4 rr 3 where the negative signs result from the inversion of the dipole moment relative to the polar axis. Substituting for M from (6), Bo(3 cos ) 2/2 IBel - (r/re) 3 2B o cos (9 (Be)r---(r/Re) 3 (10) Bo sin (9 (Be)o=- (r/re)3 From these latter equations we see that the reference field B o is simply the horizontal surface field at the CD equator (r = R e, (9 = 90ø). Most users of the centered dipole approximation to the earth's field will wish to enter the geographic coordinates of the location in question in the appropriate formulas and obtain values of the total magnetic field IBel and the magnetic field components (Be),., (Be) 0, and (Be),, (hat is, the vertical component (positive when directed outward), the geographic north-south component (positive in the direction of increasing 0, that is, when directed to the south), and the geographic east-west component (positive when directed to the east). The following procedure, based on a transform of geographic to CD coordinates, makes this possible: from 1. (5) First, and derive (6), using the dipole the chosen parameters spherical M (or harmonic Bo), 0 n and repre- Pn sentation of the earth's field. 2. Next, supposing the geographical location of the point is (r,, p), where = 90 ø-- 0 is the latitude and p the east longitude, the CD colatitude (9 of the point is computed from the expression for cos (9 in (2). For example, if the DGRF 1980 field model is used, the expression for cos (9 is cos (9 = [ cos (90 ø - ) sin (90 ø - ) cos ( p )] (11) where appropriate substitutions have been made from (7). 3. The values of (9 and radial distance r are now substitu- ted into either (9) or (10) to obtain the magnetic field quantities IBel, (Be),., and (Be) o, which apply in the CD coordinate system. 4. The field quantities IBel and (Be),. also apply in the geographic coordinate system; (Be) o does not, but it can be re- solved into the two geographicomponents (Be) o and (Be), by using (Be) o = (Be) o cos ½t (Be), = --(Be) o sin ½ ( 2) where ½t is the CD declination given by (3). 5. Finally, if required, the CD coordinates of the geographical point and the CD declination at the point can be obtained by using the expressions in (3). If it is desired to extend the above procedure to obtain CD estimates of the conventional elements of the earth's magnetic field [e.g., Parkinson, 1982; Merrill and McElhinny, 1983], the following further relations are required: F- IB I I = tan -2 [(Be),./(Be)o] I = tan-2(2 cot (9) = tan-2 (2 tan A) D=½ H = I(Be)01- I[(Be)o 2 + (Be), 212 /21 V(or Z)= --(Be),. X = --(Be) o Y = (Be), (13) where F is the total magnetic intensity (always positive), I is the inclination or magnetic dip (positive when (Be), is directed toward the earth's center), D is the magnetic variation or declination (positive when the magnetic north is to the east of true north), H is the intensity of the horizontal component of the earth's field (always positive), V (or Z) is the intensity of the vertical component of the earth's field (same sign as I), X is the north directed component of H (positive when directed to the geographic north), and Y is the east directed component of H (positive when directed to the geographic east).

5 FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES (Am 2) i i i i i I ß YEAR i I 1980 Fig. 3. Variation of the dipole magnetic moment M with time during the interval A.D. The first data point is derived from the original Gauss coefficients for the year 1835, and the final point comes from the IGRF 1985 field model Secular Change in Dipole Parameters It is interesting to make a brief historical comparison of the above 1980 and 1985 IGRF centered dipole parameters with those that follow from the Gauss coefficients derived by Gauss himself for epoch 1835 [Chapman and Bartels, 1940] and those derived by A. Schmidt, the geomagnetician who introduced geomagnetic coordinates, for epoch 1922 [Schmidt, 1934; Bartels, 1936]. First, from the coefficients derived by Gauss (epoch 1835) we obtain B o=3.31 x 10 ' nt M=8.56 x 1022Am 2 0,, = 12.2 ø ø (14) and from the coefficients derived by Schmidt (epoch 1922) we obtain B o=3.15 x 104nT M = 8.15 x 1022 A m 2 0. = 11.5 ø ø (15) Care must be taken in interpreting the changes in the properties of the centered dipole that are implied by a comparison of the data in (7), (8), (14), and (15), since the magnetic surveys on which the Gauss coefficients are based have improved greatly over the years (Gauss's data were merely adequate for a first trial of his spherical harmonic analysis, Schmidt's data depended heavily on surveys made by the wooden vessel Carnegie, and the 1980 and 1985 IGRF data benefit from satellite observations), and the changes may relate more to our improved knowledge of the magnetic field than to the secular change. However, there are consistencies to the changes, which suggest that they have some geophysical significance. To illustrate the consistency in the changes and to place the most recent changes in context, Figure 3 shows the variation of the dipole moment M with time, starting with the value given by Gauss' coefficients for epoch 1835 and ending with the value given by the IGRF 1985 coefficients. In between those two extremes the M values are also plotted for all the IGRFs and DGRFs in the interval [Barraclough, 1985], as well as the M values for 1885 and 1922 according to the spherical harmonic coefficients derived for those epochs by Schmidt [Schmidt, 1934; Chapman and Bartels, 1940]. Table 2 lists the corresponding numerical values for M. Figure 3 clearly shows the known decline of M with time [e.g., Merrill and McElhinny, 1983]. A straight line has been fitted to the points using the least squares method, and its close fit shows that the decline was essentially linear with time over the interval covered by the display. The data in Figure 3 suggest that the dipole moment will continue to decline in the near future at the rate given by the slope of the least squares fitted line, which is A m 2 per century, or roughly a 5% drop each century. However, the data in Figure 3 also show what appears to be an acceleration of the rate of decline starting around To place this accelerated decline in a more historical perspective, Figure 4 extends the time scale of Figure 3 back to the year 1600 by adding the dipole moments for eight epochs between 1600 and 1910 that result from the Gauss coefficients derived by Barraclough ] (see Table 2 for the numerical values of M; the g O coefficients in Barraclough's models for epochs before 1850 were derived by linear extrapolation from later coefficients, and thus the resulting values of M are not independent and would be expected to have a linear trend). The expanded set of points is still closely fitted by a straight line, but it now has a slope of A m 2 per century, and the apparent recent acceleration of the decline is seen more clearly to start around There is an interesting possibility that the start of the acceleration of the decline in M may relate to the magnetic "jerk" [Courtillot et al., 1978; Malin and Hodder, 1982] observed in 1970, but the relation must remain a specu- TABLE 2. CD Parameters for the Indicated Spherical Harmonic Models of the Earth's Magnetic Field Date Model x 1022 A m 2 øn øe 1985 IGRF DGRF DGRF DGRF DGRF IGRF IGRF FL IGRF IGRF Schmidt Barr Barr Schmidt Barr Gauss Barr Barr Barr Barr Barr FL denotes Finch and Leaton [1957], and Barr. is short for Barraclough [1974].

6 6 FRASER-SMITH' CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES 9,4 I I I i 9.0 m 75 ø M (Am 2) ,,:, x \ o z 70 ø./'...: }';:..'.::'?;' :,.--.-:!':;' -... i!...-.-!::i.:},: ::. X X '-'-v" YEAR I 2000 Fig. 4. Variation of the dipole magnetic moment M with time during the interval A.D. Additional points from the field models derived by Barraclough [1974] for the years A.D. have been incorporated in the data set illustrated in Figure 3. G indicates the Gauss moment for lation until further data are available; another proposed "jerk" around 1912 does not appear to have influenced the decline of the dipole moment. Figure 5 shows the positions of the north CD pole for various epochs, using the spherical harmonic data that were utilized in the preparation of Figure 3, that is, data applicable to epochs from 1835 to 1985 (see Table 2). It is commonly observed that the positions of the CD poles do not vary much with time [Merrill and Mcœ1hinny, 1983], and the data in Figure 5 clearly support that observation. On the other hand, there is some progressive variation in position, with the pole 80 ø 78 ø 76 ø 282 o 286 o 290 o 294 ø 298 o EAST LONGITUDE Fig. 5. Positions of the north CD pole for the interval A.D., using the IGRF/DGRF models for and the Barraclough [1974] models for 1890 and The position given by the original Gauss coefficients for 1835 is denoted by G. The CD pole has been located in western Greenland for well over a century, but it has now moved out into Nares Straight, which separates Ellesmere Island (on the left) from Greenland. Fig ø 270 ø 280 ø 290 ø 300 ø EAST LONGITUDE Comparison of the locations of the north CD pole and the observed dip pole, A.D. now apparently moving away from Thule in the general direction of the north geographic pole. As already noted, the exact position of the CD pole given by the original Gauss coefficients for epoch 1835 must be treated with caution, even though the value of M given by the coefficients is completely in accord with values of M derived for earlier and later epochs (Figure 3); it is shown because it is quite remarkably close to the positions given by much later field models. Figure 6 places the north CD pole data shown in Figure 5 into the appropriate geomagneti context. It will be recalled that the object of using the CD model for the earth's magnetic field is to have a simple and perhaps reasonably accurate representation for the earth's field. One test of the accuracy of the CD model is its ability to reproduce the observed "magnetic" poles, that is, the actual measured locations where the earth's magnetic field is vertical, which I will hereafter refer to as the dip poles (or observed dip poles). Figure 6 shows the north CD poles together with the actual locations of the north dip pole as recorded since its discovery near Boothia Peninsula by J. C. Ross in June 1831 [Ross, 1834]. The coordinates are taken from Dawson and Newitt [1982] and include the observation by Amundsen, the observation by Serson and Clark, the observation by Dawson and Loomer, and the observation by Niblett and Charbonneau. To these I have added the position reported by Canadian scientists [see Newitt, 1985; Newitt and Niblett, 1986]. It is clear from the data in Figure 6 that the CD pole is only a very crude approximation to the observed dip pole. Figure 7 shows the shift of the south CD pole from 1850 until the present. There are no significant geographical features in the vicinity of the CD pole except for the Soviet station ¾ostok ( ø, ø). Considering its motion relative to ¾ostok, the CD pole appears to have moved in such a way as to reduce its distance from the station until about 1975, but it is now beginning to move away in the general direction of the south geographic pole. 3. ECCENTRIC DIPOLE 3.1. Derivation of the ED Position Coordinates Once it is decided to approximate the earth's field by a magnetic dipole not necessarily located at the geographic

7 FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES 7 / 102o /--- _ '-'- / 76øS 78øS I 106o-----Ir 1985 I + +-t- I + 110OE-- _ 1945-H' 191e J / ø / / +G / t / '"-- 118ø- -._ /- Fig. 7. The southern equivalent of Figure 5, showing the south CD pole positions in Antarctica for the interval A.D., using the same field models as were used for Figure 5. The location of the Soviet Antarctic station ¾ostok (¾O) is also shown. where B o is the reference field defined in (5), and L o = 2gløg2 ø q-(3)l/2[gllg21 q- hllh21] L 1 = --gllg2 ø + (3)l/2[gløg21 + gllg22 + hllh22] (17) L 2 = --hllg2 ø q- (3)l/2[gløh21 -- h lg2 2 q- gllh2 2] E = (Logl ø + L gi + L2hll)/4Bo 2 The total shift of the dipole from the center is g, given by ( 2 q- r/2 q- 2)l/2RE (18) It follows from the above equations that the eccentric dipole is completely specified by first eight Gauss coefficients. Substituting the values of the 1980 Gauss coefficients (Table 1) in (17), the following numerical values are obtained: L o=8.887 x 107nT 2 L x = x 108nT 2 L 2 = x 108 nt 2 E = x 102 nt which give r/= , = , and = The shifts Ax, Ay, and Az in the x, y, and z coordinate directions are therefore km, km, and km, respeccenter, the question then arises as to what criterion is to be tively, and the total distance shifted by the dipole is 6 = used to judge the best fit to the observed field. The criterion km. The direction of the shift is given by adopted by Schmidt [1934] and described by Barrels [1936] is cos- x (170.2/488.6) = 69.61ø, and pd = 90 ø + tan- I (385.4/ to minimize the terms of second order in the potential used in 247.5) ø, that is, it is toward the point 20.39øN, the spherical harmonic representation of the field. The eccen øE. This point is in the northwest Pacific, at the northern tric dipole so obtained has the same moment as the centered end of the Mariana Islands. dipole and the same orientation of its axis, but in terms of the geographic rectangular coordinate system x, y, z (Figure 1) it 3.2. Secular Change in ED Position is located at a position Ax = r/re, Ay = Re, Az = Re, where If the IGRF 1985 Gauss coefficients are substituted in (17), the quantities r/,, can be derived from the Gauss coefthe position parameters for the eccentric dipole are found to ficients, as described in the following paragraph. It might be be Ax = km, Ay = km, Az = km, and noted at this point that the rectangular coordinate designa km. This result suggests that the dipole is moving tions used by Schmidt [1934] and Barrels [1936] differ from away from the earth's center. Indeed, computations with the those now conventionally used, for example, the x axis is used complete set of!grf and DGRF data for the interval for what is now conventionally the z axis, and in the work by 1985 indicate that the dipole has been gradually drifting away Chapman and Barrels [1940] this circumstance has led to an from the earth's center since To put this drift into pererroneous designation of the shifts Ax, Ay, and Az. When spective, I have compute d the position of the dipole since reference is made to the eccentric dipole model of the earth's 1600, using the same Gauss coefficient data sets that were used magnetic field, it is now generally understood that the Schmidt to investigate the secular variation of the dipole moment M in [1934] criterion and its resulting mathematical' formulation section 2.4 (note that the same secular variation of dipole are applicable, e..ven though other eccentric dipole models are moment applies in the case of the eccentric dipole, since the possible [e.g., Bochev, 1969a], and it is the Schmidt eccentric ED and CD moments are identical). The ED position data dipole model that is described in this work. There is not an obtained from these computations, together with the correextensive literature treating the eccentric dipole formalism; the sponding distance 6 from the earth's center, are listed in Table major works are those by Schmidt [1934], Barrels [1936], and 3, and the results are illustrated in Figures 8, 9, and 10. Chapman and Barrels [1940], together with valuable contri- Figure 8 shows the secular variation of the distance 6 of the butions by Akasofu and Chapman [1972], Ben'kot, a et al. eccentric dipole from the earth's center. There appear to be [1964] and James and Winch [1967]. Other relevant articles three different regimes over the time interval covered by the include those by Vestine [1953], Parkinson and Cleary [1958], display'(1) a steady decline of 6 throughout the interval Cole [1963], Kahle et al. [1969], Parkinson [1982], and Wallis 1800, (2) a steady increase from 1800 to around 1920, and (3) et al. [1982]. ' an accelerated steady increase from 1920 until the present. As The dimensionless coordinate quantities, r/, and are can be seen, the eccentric dipole is now farther from the given by earth's center than it has been at any other time since at least 1600; at roughly 500 km the distance i s about 7.8% of the = (L o -- gløe)/3bo 2 earth's radius. On the basis of its recent trend we can expect r/= (L1-- gl 1E)/3Bo 2 (16) the distance 6 to continue increasing in the near furture at what appears to be an historically substantial rate. Thus the (L 2 _ hi 1E)/3Bo 2 distance, already nearly twice its average value during the (19)

8 FRASER-SMITH' CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES TABLE 3. ED Position Coordinates, as Measured in the Geographic Rectangular Coordinate System, and the Distance g of the Dipole From the Earth's Center for the Indicated Spherical Harmonic Models of the Earth's Magnetic Field Ax, Ay, Az, g, Date Model km km km km 200 loo I 1985 IGRF DGRF DGRF DGRF DGRF IGRF IGRF FL IGRF IGRF Schmidt Barr Barr Schmidt Barr Gauss Barr Barr Barr Barr Barr FL denotes Finch and Leaton [1957], and Barr. is short for Barraclough [ 1974]. interval , should continue to set new records for some years to come. The 1955 Finch and Leaton magnetic field model is included in the ED computations reported here and in the previous CD computations (see Table 2) to provide a check against the results of Parkinson and Cleary [1958], who used the Finch and Leaton model. Comparing the results for 5, Parkinson and Cleary report a value of "about 436 km" as compared with g km in Table 3. The displacement of the dipole is toward a point at 15.6øN, 150.9øE according to Parkinson 5OO ß 200 I 1600 ß Ge I I I : YEAR 2000 Fig. 8. Variation of the distanc e 6 of the eccentric dipole from the earth's center over the interval A.D. The distance implied by the original Gauss coefficients for 1835 is denoted by G. Three regimes have been indicated by straight lines. The IGRF/DGRF models for and the field models of Barraclough [1974] for were used primarily for this display. -loo - -2oo i i i 16oo 17oo 18oo 19oo 2000 YEAR Fig. 9. VariatiOn of the distance Az of the eccentric dipole above the geographic equatorial plane during the interval A.D. As can be seen, the dipole was located below the plane for much of the interval, but it is now at its greatest distance above the plane for the last four centuries. The IGRF/DGRF models for and the field models of Barraclough [1974] for were used for this display. and Cleary, while the data in Table 3 imply a displacement toward 15.7øN, 150.8øE. The agreement between these numbers is close; further, it is as close as might be expected, since the numerical values for the dipole moment and ED coordinates depend on the numerical value that is chosen for the earth's radius and Parkinson and Cleary do not document the precise value used in their work. Figure 9 shows the variation of the distance Az, that is, the distance of the eccentric dipole above the geographic equatorial plane, since For much of the interval the dipole has been below the equatorial plane, but it moved above the plane around the end of last century, and it is now at its largest distance above the plane. On the basis of the trend since 1900 it can be expected to move to new record distances above the plane over the next few decades. Finally, Figure 10 shows the variation since 1600 of the point of projection of the eccentric dipole position on the geographic equatorial plane. The data prior to 1800 do not show any steady trend, but the point of projection appears to have been moving steadily toward the western Pacific for roughly the last 200 years ED Axial Poles The eccentric dipole model for the earth's magnetic field produces two different varieties of poles. The first of these are what I will refer to as the axial poles (the two points on the earth's surface where the ED axis intersects the surface). Because of the displacement of the eccentric dipole away from the earth's center the ED axis and the ED magnetic field, in particular, are not perpendicular to the surface at the ED axial poles. There are, however, two points where the ED magnetic field is perpendicular to the surface, and I will refer to these points as the ED dip poles. In this section, expressions will be derived for the positions of the axial poles. We know that the axis of the eccentric dipole is parallel to the CD axis. This fact and the knowledge that the ED axis passes through the point (Ax, Ay, Az) in geographic rectangular coordinates enable us to derive an expression for the ED

9 FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES ß z !985 ß J ,,._ 1975,ßJß ' 18y1922 -loo 1850 I ' ' - '- y (km) t x (kin) ( :o o) Fig. 10. Variation of the projection of the eccentric dipole's position on the geographic equatorial plane during the interval A.D. Once again the IGRF/DGRF models for and the field models of Barraclough [1974] for were used primarily for the display. axis in our basic geographic spherical polar coordinate system. The ED axial poles are then found comparatively simply by finding the points of intersection of the axis with the surface r - R e, representing the earth. The equation for a line passing through a point (Ax, Ay, Az) in the geographic rectangular coordinate system is x - Ax y- A y z - Az - - (20) I m n where 1, m, and n are the direction cosines of the line. Converting to geographic spherical polar coordinates and substituting 1= sin 0. cos b., m = sin 0. sin b., and n = cos 0., which follow from the known orientation of the dipole axis in geographic coordinates (Figure 1), we obtain r sin 0 cos b - Ax r sin 0 sin b- Ay r cos 0- Az = = (21) sin 0. cos 4. sin 0. sin b. cos 0. as the equation of the ED axis. Substituting r = R e in (21) and carrying out the appropriate algebraic manipulations, the following equations are obtained for the points of intersection of the ED axis with the earth's surface, that is, for the ED axial poles: L(Re- [(Re -- Az) Az) cos sin 4. b. tan tan AY x 1 (22a) O = sin-' [.Ax ck"- Ay cøs ck" (22b) R e sin ( b.-- b) b = tan-' for the north ED axial pole, and b = tan-' [ L(Re ('Re + Az)cos sin b. tan A x y] (23a) O=180ø-sin-Z[ Axsinck"--Aycøsck".] R e ( b. b),23b) for the south ED axial pole. To use these equations to derive the locations of the poles, the quantities Ax, Ay, Az, 0., 4., and R e are first substituted in (22a) and (23a) to obtain the 4 values for the two poles. These 4 values, together with the given data, can then be used in the appropriate expression for 0 to obtain the polar colatitudes Recent Positions of the ED Axial Poles and Their Secular Change Table 4 lists the computed ED axial pole positions for all the IGRF and DGRF field models and for the original Gauss coefficients for epoch In addition, for comparison with the results of Parkinson and Cleary [1958], the pole positions are also listed for the spherical harmonic field model of Finch and Leaton [1957] for epoch 1955, the field model used by Parkinson and Cleary [1958]. Finally, to provide information about their likely change over the next few decades, the pole positions are tabulated for the earlier field models of Schmidt [Schmidt, 1934' Chapman and Bartels, 1940] and Barraclough [1974]. Comparing the results of the axial pole computations for the Finch and Leaton [1957] field model for epoch 1955 with the results obtained for the same model by Parkinson and Cleary [1958], Table 4 shows the north ED axial pole at 80.90øN, 275.6øE, whereas Parkinson and Cleary obtained 81.0øN, 275.3øE (84.7øW). There is similar close agreement for the south poles. The north ED axial pole is currently located in the sea just off the northwest coast of Ellesmere Island, in the Canadian Arctic. Figure 11 shows its 1985 position, together with previous positions back to the year It has moved over a greater distance in the time interval than has the CD pole. The south ED axial pole is now located in a remote part of the Antarctic continent, as shown in Figure 12. It is about 400 km from Vostok along the line joining Vostok with Porpoise Bay. Interestingly, it was probably located on the Ross Ice Shelf prior to ED Dip Poles As pointed out in section 3.3, the ED magnetic field is not perpendicular to the earth's surface at the ED axial poles, due to the offset of the dipole from the earth's center. However, there are two points where the ED magnetic field is perpendicular to the surface. One of these points is near the north ED axial pole and the other near the south ED axial pole; they will be called the north and south ED dip poles, respectively. Both dip poles are located on the great circle defined by the intersection of the plane containing both the CD and ED axes with the earth's surface; they are separated from their corresponding ED axial poles by small angular distances along the great circle, with the direction of the separation being away from the local CD pole (the CD and ED poles all lie on the great circle). At these points there is enough curvature of the dipole field lines away from the ED axis to compensate for the small angle made by the axis with the earth's surface and thus to bring the field lines perpendicular to the surface (Figure 13a). It is not difficult to compute the geographic locations of the ED dip poles, but the computations are involved, and ultimately, as we will see, the equation for the pole positions must be solved numerically. The procedure that was used here con- sists of the following several steps: In the first step a transform is made into the CD coordinate system. The only significant feature of this step is a change of

10 10 FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES TABLE 4. Geographic Coordinates of the Axial and Dip Pole Positions for the Eccentric Dipoles Resulting From the Indicated Spherical Harmonic Models of the Earth's Magnetic Field Axial Dip North South North South Date Model Latitude, Longitude, Latitude, Longitude, Latitude, Longitude, Latitude, Longitude, deg øe deg øe deg øe deg øe 1985 IGRF DGRF i DGRF ! !970 DGRF DGRF IGRF IGRF FL IGRF IGRF Schmidt Barr Barr Schmidt Barr Gauss Barr Barr Barr Barr Barr ! FL denotes Finch and Leaton [1957], and Barr. is short for Barraclough [1974]. the coordinates of the eccentric dipole. To operate in the CD system, it is necessary for the position coordinates of the eccentric dipole to be converted to their appropriate CD coordinate system form. Thus instead of the geographic rectangular coordinates (Ax, Ay, Az) or equivalent geographic spherical polar coordinates (6, 0d, 4 d), where 6 = (Ax 2 + Ay 2 + Az2) /2 0, = 90 ø -- 3,, = cos- (Az16) t24) ba = tan- x (Ay/Ax) the eccentric dipole now has the CD position coordinates (AX, A Y, AZ) and (6, O a, d), where uppercase is used, as before, to EAST LONGITUDE Fig. 11. Positions of the north ED axial pole for the interval A.D., using the IGRF/DGRF models for and the Barraclough [1974] models for Ellesmere Island is in the center of the display, and Greenland is to the right. G is the ED axial pole position given by the original Gauss coefficients for Resolute Bay is denoted by RE. Fig. 12. The southern equivalent of Figure 11, showing the positions of the south ED axial pole (solid squares) for the interval A.D., using the IGRF/DGRF models for and the Barraclough [1974] models for Also shown are the CD pole positions (crosses) for the same interval. In addition to Vostok (VO) the positions of the Soviet station Mirny (MI), the Australian station Casey (CA), the French station Dumont D'Urville (DU), and the U.S. station McMurdo (MM) are shown.

11 FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES 11 13b. The dip pole condition is then Bre sin ( -- 0e) -- Boe cos ( -- 0e) = 0 (27) which, after substitution for Br and Bo, becomes 2 tan ( - 0e) = tan 0 e (28) This equation now has to be solved for. Applying the sine rule to the triangle OEP, we have a Fig. 13. Condition for the occurrence of an ED pole. Both panels of this figure show part of the plane defined by the CD and ED axes and its intersection with the earth's surface, represented here by a segment of a circle. O is the earth's center, E is the location of the eccentric dipole, OB is part of the CD axis with B the north CD pole, DEA is part of the ED axis with A the north ED axial pole. (a) The magnetic condition for the occurrence of the north ED dip pole is illustrated (the figure is not drawn to scale). One of the dipole field lines is perpendicular to the earth's surface. (b) The geometry that is used in the derivation of the equation for the dip poles. P is a general point in this panel and not necessarily the point where the dipole field is perpendicular to the surface. b r e 6 R e = = (29) cos( +Aa) sin( -0e) cos(0 e+aa) Further, if F is the foot of the perpendicular from E onto the line OP, we have R e - OF + FP, which gives R e = 6 sin (O + Aa) + r e cos (O -- 0e) (30) From (29) and (30) we can write 6 cos (O + Ad) sin (O -- 0e) = r e R E -- 6 sin (O + Aa) cos (!9-0e) = r e (31) giving distinguish the specifically CD quantities. Equations analogous to (24) also apply for the CD position coordinates of the dipole' g--(ax 2 3. A Y 2 3- AZ2) 1/2 O d = 90 ø - A a = cos- 1 (AZ/6) (25) tan- 1 (A Y/AX) The change of position coordinates from (6, 0 (1)a) is easily carried out by using the procedures detailed in section 2. To illustrate this particular change of position coordinates, let us take the IGRF 1985 field model as an example. In geographic rectangular coordinates the eccentric dipole is located at ( km, km, km), as shown in Table 3. The equivalent geographic spherical polar coordinates are (502.0 km, ø, 146.7ø). In the CD coordinate system the rectangular coordinates are ( kin, kin, km), and the polar coordinates are (502.0 km, ø, 215.6ø). The second step in the derivation is to obtain the CD coordinates of the ED dip poles. Figure 13b shows the geometry required in this step of the derivation, that is, the same plane is involved as in Figure 13a, and it follows that the CD azimuthal coordinates for the two ED dip poles are the same and equal to ti) a' the azimuthal angle does not play a further role at this stage of the derivation. The other CD coordinates of the ED dip poles are now obtained by resolving the magnetic field of the eccentric dipole, located at E in Figure 13b, along the tangent to the circle (representing the earth's surface) at the general point P, equating the resolved field to zero, and then rearranging the resulting equations to obtain an expression for. Remembering that the dipole at E is oriented along the axis DEA in Figure 13b, the two components of the dipole field at P are 2poM cos 0 4tOre 3 laom sin O 4rcr e 3 (26) where r e and 0 e are the ED polar coordinates of P in Figure 5 cos (19 + Aa) tan (0 -- 0e) = (32) R e --6 sin ( + Aa) The point P in Figure 13b has CD coordinates (R e, ) and ED coordinates (%, 0e) which, from the geometry of Figure 13b, implies the following two results: giving R e cos 19 = AZ + r e cos 0 e R e sin = (AX 2 + Ay2)l/2 + re sin 0 e (33) R e sin cos A a tan 0 e = (34) R e cos sin A a Substituting the expressions for tan (0-0e) and tan 0 e, given by (32) and (34), into the dip condition (28), and carrying out the necessary algebraic manipulation, the following expression for 19 is obtained: COS 2 ( }- K cos 19 sin 19- K 2 cos O-- K 3 sin = 0 where K = tan A a (35) K 2 = 36 sin Ad/R e (36) (Re ) sin 2 A a K 3 -- fir e cos A a Equation (35) must be solved numerically for, and being of the second order in cos O, it gives two values, O x and 0 2, corresponding to the north and south ED dip poles. The third and final step in the derivation is to convert the CD coordinates (R e, ( }1,2' Od) of the ED dip poles into geographic coordinates using the procedures described in section Recent Positions of the ED Dip Poles and Their Secular Change Table 4 lists the computed ED dip pole positions for the same field models that were used to obtain the ED axial pole

12 12 FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES o =1985, ß = FL, ß = 1955, o= øS o J \\?;:: ø CD Poles I ED.,xia 200 / / EDDip :- - ' // Poles ::'/ 130 Fig. ]4. The Antarctic positions of the CD and ED axial and dip poles for the Fi c a L ato []957] field model used n Cleary []958]. Also shown arc the sam pole positions for the ]945, ]955, and ]985 ]GEF [ ld models. Note how the three varieties of coma nctic poles arc approximately colinca in this map projection. positions (section 3.4). Once again comparing the results obtained for the Finch and Leaton [1957] model for epoch 1955 with those of Parkinson and Cleary [1958] for the same model, Table 4 shows the north ED dip pole at 82.46øN, 222.0øE, whereas Parkinson and Cleary obtained 82.4øN, 222.7øE (137.3øW). There is therefore close agreement between the computed coordinates for this pole and similar close agreement between those for the south dip pole. Figure 14 shows the Antarctic positions of the CD and ED axial and dip poles for the Finch and Leaton [1957] field model as well as the same pole positions for the 1945, 1955, and 1985 IGRF field models. One purpose of Figure 14 is to show the magnitude and direction of the shift of the poles away from the geographic south pole as the pole models are progressively refined from centered dipole to eccentric dipole (axial pole) to eccentric dipole (dip pole). It does not appear to be generally recognized that in each polar region the CD, ED axial, and ED dip poles derived from a particular field model lie approximately along a straight line, depending on the map projection that is used, a result that follows immediately from their colocation on the great circle segment shown in Figure 13. It also follows that the geographic poles are not in general aligned, even approximately, with their respective triad of magnetic poles. Another purpose is to show the comparatively small movements of all the magnetic poles in the interval The ED dip poles have moved the most, with the 1985 pole located in the waters of Porpoise Bay in Wilkes Land, whereas the 1945 pole was well inland. Finally, com- parison of the IGRF 1955 and Finch and Leaton pole positions gives an idea of the variability associated with two different models for the same epoch. Figures 15 and 16 show the positions of the north and south ED dip poles for various epochs in the interval A.D., and they summarize the various CD and ED pole positions and their motion since The CD, ED axial, and ED dip poles in these figures were computed solely from the IGRF and Barraclou /h [1974] field models, and the numerical data are listed in Tables 2 and 4. There is just one exception, the ED dip pole positions that follow from the original Gauss coefficients for epoch 1835, which are included for continuity with the earlier displays. The pole positions for the 1600 A.D. field model have been excluded because they are less reliable than the others (Barraclou /h [1974] lists significantly larger standard deviations for the Gauss coefficients), and unlike the other positions they do not always conform 75 ø,_ + = CDpole, ß= ED axial pole, ß= EDdippole o e-- '1 80 o, 85,. - -'n / -"t'".. / / :. -0 AN I..:... :::,.. OCE / z,, / 'ox / ' 1650: ' ":: :,. Z 70 ø, :.: :. : G. : J9 5 TM 'X X GREENLAND.... > 240 ø ':.., ' 1831 t: :? ': %.:. %,. 32(), 250 ø 260 ø 270 ø 280 ø 290 ø 300 ø 310 ø EAST LONGITUDE Fig. 15. Summary of the CD, ED axial, ED dip, and measuredip pole positions for the northern polar region. All the poles are currently moving roughly to the north, or northwest, as indicated by arrows for the ED axial and dip poles. The measuredip pole positions, denoted by small solid circles, are the same as those shown in Figure 6, and are described in the associated text.

13 FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES 13 + = CDPole.= ED axial pole ß= EDdippole / i' %:. /' \ / / 70øS '% ii..x / 60øS / / / / x : :".::_,.. ' ;' ":.:i.-.'.. \\ 90 o E : OE 1985' CA :::.. /+ I.:?' + +. s I. ' 1650 / /?' / : :: ' 1985./ I '1650. :':: / MM? G //1962. :' L / :t::'t; ' '::':' ' / ' 130 o E Fig. ]6. Summar of the CD, ED axial, ED dip, and measured dip pole positions for the southern polar region. The arrows indicate the apparent present direction of motion of the poles. well to the trend established by the positions for neighboring epochs. Also shown in both Figures 15 and 16 are a number of the measured (or inferred) dip pole positions (usually called "magnetic poles") that have been obtained as the result of various expeditions. Needless to say, one of the tests of a model of the earth's magnetic field is how well it predicts the actual observed locations of the north and south dip poles. In both Figures 15 and 16 it is evident that the ED dip pole positions are in much better agreement with the observed dip pole positions than are the CD pole positions. Considering the individual figures, in Figure 15 we see that the paths followed by the ED dip pole and the measured dip pole have been roughly parallel since the beginning of the century and that they should continue to be parallel for some time into the future. Around there was a comparatively very abrupt change in the direction of the path being followed by the ED dip pole. The change was so abrupt that it appeared possible that the computations of the pole positions were in error, but no error could be found. Confirmation of the correctness of the results was obtained by applying the "straight-line criterion" for the triad of geomagnetic poles: despite the change in path, the CD, ED axial, and ED dip poles continued to be closely colinear. Since the computations for the three different poles are independent, the abrupt change must be considered a genuine feature of the ED dip pole motion. There is no such feature in the motion of the ED dip pole in the south. The measured dip pole positions in Figure 16 come from a variety of sources, including Dawson and Newitt [1982] in particular. Some emphasis has been given to relatively localized measurements, and thus many of the early pole positions inferred solely from field measurements at sea off the Antarctic coast have not been included. The first position, for epoch , is 72.9øS, 156.4øE, and it comes from measurements made during Scott's first expedition, the British Discovery expedition of [Bernacchi, 1908]. The second and third positions com from measurements made by parties that attempted to reach the dip pole: (1) the David, Mawson, and MacKay party of the British Antarctic Expedition of , which obtained a position of 72.42øS, 155.3øE for epoch [Fart, 1944] that was later corrected to 71.6øS, 152.0øE [Webb, 1925] and (2) the Bage, Webb, and Hurley party of the Australasian Antarctic Expedition of , which obtained a position of 71.17øS, 150.8øE for epoch [Webb, 1925]. The fourth position is inferred from the measurements of Kennedy during the British, Australian, and New Zealand Antarctic Research Expedition of ; it is 70.3øS, 149.0øE for epoch [Fart, 1944]. The fifth position was measured by Mayaud [1953a, b] during the French South Polar Expedition of : 68.10øS, 143.0øE for epoch The sixth position, at 67.5øS, 140.0øE for epoch , was obtained by Burrows and Hanley [Burrows, 1963]. This appears to have been the last measurement of the dip pole on land. Already close to the sea during , it must have moved out to sea around There appear to have been no documented attempts in the interval to measure the location of the dip pole directly. On January 6, 1986, scientists from the Australian Bureau of Mineral Resources aboard the M/V Icebird located the pole at 65.3øS, 140.0øE [Barton, 1986]. One of the interesting features of both Figures 15 and 16 is the comparatively abrupt recent change in the direction of motion of the CD pole. In the south it is now moving away from Vostok, whereas a few decades ago it was moving toward the station. None of the other southern geomagnetic poles show any comparable change in their paths. In fact, for the map projection that is used, their paths are remarkably straight. As noted above, there is a considerable distance (about 400 km) between the ED axial pole and Vostok. From the point of view of phenomena in the upper atmosphere that relate to the dipole part of the earth's magnetic field, Vostok is not ideally located at its present position near the CD pole, since the dipole axis (the axis of symmetry) is more accurately that of the eccentric dipole, which intersects Antarctica at the ED axial pole. The small tilt of the ED axis relative to the earth's surface is not significant in this context, since the CD and ED axes are parallel, and the tilt at the ED axial pole is merely the result of the earth's curvature. For this somewhat negative reason the motion of the CD pole away from Vostok is unlikely to have implications for upper atmosphere physics CD/ED Coordinate Transforms and ED Magnetic Fields Cole [1963] has provided the necessary details for transforming from geographic to eccentric dipole coordinates, and he has, in addition, given plots of ED latitude and longitude for epoch 1955 (using the Finch and Leaton [1957] field model) superimposed on three geographic grids: one for the world, using a Mercator projection, and two covering each of the polar regions. A more general treatment of transforms between geophysical Cartesian coordinate systems, but one that does not specifically include the eccentric dipole system has been given by Russell [1971]. More recently, Wallis et al. [1982] derived ED coordinates for the presentation of Magsat data. In view of this earlier work I will, in this section, present only the transform equations necessary for conversion between the CD and ED systems. The transform between CD and ED coordinates is straightforward, since only a translation is involved: The origin of the ED system is located at the point (AX, A Y, AZ) in the CD system, but the directions of the Cartesian axes are the same.

14 14 FRASER-SMITH' CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLEG Using this information, the basic equations relating the CD and ED coordinates are X = r sin O cos ß = x e + AX = F e sin 0 e COS e q- AX Y -- r sin O sin b - Ye + AY -- r e sin 0 e sin 4 e + AY (37) Z F cos { z e q- AZ r e cos 0 e q- AZ where the subscript e is used to denote ED coordinates X, Y, Z' r,, and ß are the Cartesian and spherical polar coordinates in the CD system, and Xe, Ye, Ze; re, Oe, and Pe are the corresponding coordinates in the ED system. The radial coordinate r is common to both the geographic and CD systems. From these equations we obtain rsinocos --AX giving sin 0e COS (pc rsin sin --AY sin 0e sin pe r cos 0 - AZ = (--re) (38) COS 0 e - [ sin sin ß- sin O cos ß- A X ] (39a) tan- I (; r COS sin ( O -- c os AZ) _AX_ cos 4e] 1 (390) 4 e = tan 0e = To transform from CD to ED coordinates, Pe is computed from (39a), followed by 0 e from (390) and r e from r cos O - r e = cos 0 e AZ (39c) which follows from (38). In the event that (Pc-- 90ø or 270 ø the equation to use for 0 e is Oe=tan- [ rsinosinrd--ay] r cos O- AZ (40) Similarly, if 0 e = 90 ø or 270 ø, the equation to use for r e is r e = [(r sin O cos ß -- AX) 2 + (r sin O sin ß - AY)2] 1/2 (41) To transform from ED to CD coordinates, the required equations are ß =tan r e sin0 esin pe+ ' sin 0 e cos Pe + (42a) [r( esinoecos 0 e q- AZ) CPeq-AX g 3] O = tan- (420) r e cos 0e + AZ r = cos O (42c) with equations similar to (40) and (41) when ½D and O are 90 ø or 270ø: O=tan- [ resinoesinqbe+ay] recos 0e+AZ (43) r = [(r e sin 0 e cos Pe + AX) 2 + (re sin 0 e sin Pe + Ay)2] 1/2 (44) Of these two transforms it is likely that the one from CD to ED coordinates will be the most useful, since it is the one required to compute the ED magnetic field at a given geographic location. Considering the ED magnetic field, it will be given, in the ED coordinate system, by #om(3 cos 2 0 e q- 1) /2 4Itr e 3 2#oM cos 0e (BE)re = 47ire 3 (45) #o M sin 0 e (BE)Oe : 471;re 3 Suppose now that the total ED magnetic field is required at a particular geographic location. To obtain the field, the dipole moment M and north CD pole position coordinates 0, and p, are computed from the spherical harmonic field model of choice, using the procedure detailed in section 2.2. Next, the ED Cartesian position coordinates are computed from the field model using the equations in section 3.1; these coordinates should immediately be converted to their CD form as described in section 3.5. The final preparatory step is to con- vert the geographic coordinates of the location of interest to ED coordinates via an intermediate conversion to CD coordi- nates; the procedures are described in section 2.1 (geographic to CD) and in this section (CD to ED). With the ED coordinates of the point and the scalar moment M it is then possible to calculate the total magnetic field from the equation for IB l in (45). A similar but more involved procedure is required to obtain the ED magnetic field components. 4. DISCUSSION The principal results of this work are the pole positions and other dipole data presented in the various figures and tables in the text. With the exception of the observed dip pole positions, which provide a check on the applicability of the dipole field models, the data were all computed, with negligible error, from the original Gauss coefficient field models. Thus the accuracy of the pole positions relates directly to the accuracy of the Gauss coefficients. However, this is not an important point at the presen time, since only the dip pole positions appear to be accurately measurable quantities. A general question, arising out of this work, concerns the fidelity with which the dipole field models represent the actual field of the earth. As is shown by the summary figures, Figures 15 and 16, the computed ED dip pole positions are much closer to the observed dip pole positions than are the computed CD poles. From this point of view the ED model pro- vides a superior representation of the earth's field, as compared with the CD, or "geomagnetic," model. However, the ED dip poles do not closely correspond to the observed dip pole positions. Better agreement may be obtained by using the complete spherical harmonic field models for each epoch (that is, by using all the Gauss coefficients) to compute the dip pole positions, or even by combining a number of field models to produce an average spherical harmonic model for each epoch and then computing the dip pole positions, as was done by Dawson and Newitt [1982]. Even with the best modeling, however, there are still discrepancies between the observed and computed fields. Under these circumstances it would be helpful to have a quantitative measure of the agreement between a particular geomagnetic field model and what I will refer to as

15 FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES the measured field, that is, a set of geomagnetic field measurements that, perhaps by general agreement, are sufficiently timely, accurate, and complete to be used as the basis for a model. For example, the set of measurements on which the world magnetic charts and their associated spherical harmonic models were based [Barker et al., 1981]. In the following, interpolation between the measured values may be required to give the necessary worldwide coverage. This measure, a goodness of fit index (GFI), would enable a better informed judgment to be made concerning the use of a particular field model' Under some circumstances the centered dipole model may be entirely adequate; under other circumstances the use of the presumably more accurate eccentric dipole model would be desirable or, if high accuracy was required, the GFI would enable the most accurate spherical harmonic model to be selected. One possible GFI could be constructed as follows. A particular field quantity, the total field for example, is computed from the model for each 10 ø intersection of the latitude and longitude lines, starting with the equatorial point on the prime meridian (that is, with the point 0øN, 0øE) and including the geographic poles. The magnitude of the percentage difference between the computed value and the measured value of the field quantity at each point is determined and the median value, or alternatively the average value, of the magnitudes for all 614 points computed. This final computed value would then serve as the index for goodness of fit. Because of the more rapid decline with distance of the higher order terms in the spherical harmonic field models and corresponding change in the actual measured field of the earth the GFI for any field model will be a function of altitude, and the GFIs for the centered and eccentric dipole models will tend to approach those for their originating spherical harmonic models as the altitude is increased. But why bother using a centered dipole or eccentric dipole model for the earth's magnetic field when, with modern computers, it is nearly as easy and fast to use a full spherical harmonic field model? The answer is because of the combi- nation of geophysical insight and adequate accuracy for many purposes provided by the dipole models. To give one example, the motion of charged particles in a magnetic dipole field has been the subject of much study and is reasonably well understood. Thus general statements can be made about the expected motions of charged particles in the earth's magnetic field simply by assuming that it can be represented by a dipole field. There seems little doubt that t he eccentric dipole representation of the earth's field is closer to the real field than the centered dipole representation, and one of the purposes of this paper is to make the superior eccentric dipole representation more accessible. However, it must also be pointed out that the eccentric dipole representation itself has the potential for improvement. The off-center dipole of Bochev [1969a], for example, where the orientation of the dipole axis is no longer restricted to that of the centered dipole, could conceivably give a marginally better approximation to the earth's field. More substantial improvements would be expected from the inclusion of additional dipole sources [e.g.,bochev, 1969b] or, more generally, from the use of a multipole representation of the field [Umow, 1904' Winch and Slaucitajs, 1966a, b' Zolotoy, 1966' James, 1967, 1968, 1969' Winch, 1968' Winch and Malin, 1969]. The conventional spherical harmonic representation of the earth'g field is already essentially in the required Legendre polynomial form [e.g., Stratton, 1941] for a general multipole representation of the field, and as shown by James [1967, 1968] and Winch [1968], for example, the strengths and axial directions of the geographically centered multipoles can be computed from the Gauss coefficients. The centered dipole representation follows identically from the first-order terms of the multipole expansion. However, the next higher order of the multipole expansion, to magnetic quadrupoles, does not lead to the eccentric dipole representation, although the magnitude and direction of the eccentric dipole's displacement from the earth's center can be related to the parameters of the geomagnetic quadrupole [Winch and Slaucitajs, 1966b; Winch, 1968]. Since the eccentric dipole is located at a finite distance from the earth's center and the multipoles of all orders are located at the center, it would appear unlikely that the eccentric dipole field model could be reproduced as a special case of the multipole representation. However, since both representations can be derived from the one basic spherical harmonic field model, a relationship is implied. Whatever this relationship may be, the two representations provide different views of the earth's magnetic field, and their relative merits depend on such practical factors as their accuracy and the physical insight they give. 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