Inclination Flattening and the Geocentric Axial Dipole Hypothesis

Earth and Planetary Science Letters 233 (2005) 247–261 www.elsevier.com/locate/epsl Frontiers Inclination flattening and the geocentric axial dipole hypothesis Lisa Tauxe Scripps Institution of Oceanography, La Jolla, CA 92093-0220, USA Received 26 April 2004; received in revised form 25 January 2005; accepted 31 January 2005 Available online 23 March 2005 Editor: A.N. Halliday Abstract William Gilbert first articulated what has come to be known as the geocentric axial dipole hypothesis. The GAD hypothesis is the principle on which paleogeographic reconstructions rely to constrain paleolatitude. For decades, there have been calls for permanent non-dipole contributions to the time-averaged field. Recently, these have demanded large contributions of the axial octupole, which, if valid, would call into question the general utility of the GAD hypothesis. In the process of geological recording of the geomagnetic field, bEarth filtersQ distort the directions. Many processes, for example, sedimentary inclination flattening and random tilting, can lead to a net shallowing of the observed direction. Therefore, inclinations that are shallower than expected from GAD can be explained by recording biases, northward transport, or non- dipole geomagnetic fields. Using paleomagnetic data from the last 5 million years from well-constrained lava flow data allows the construction of a statistical geomagnetic field model. Such a model can predict not only the average expected direction for a given latitude, but also the shape of the distribution of directions produced by secular variation. The elongation of predicted directions varies as a function of latitude (from significantly elongate in the up/down direction at the equator to circularly symmetric at the poles). Sedimentary inclination flattening also works in a predictable manner producing elongations that are stretched side to side and the degree of flattening depending on the inclination of the applied field and a bflattening factorQ f. The twin tools of the predicted elongation/inclination relationship characteristic of the geomagnetic field for the past 5 million years and the distortion of the directions predicted from sedimentary inclination flattening allows us to find the flattening factor that yields corrected directions with an elongation and average inclination consistent with the statistical field model. The method can be tested using sediments deposited in a known field. Application of the elongation/inclination correction method to two magnetostratigraphic data sets from red beds in Asia and Pakistan brings the inclinations into agreement with those predicted from modern GPS measurements and from global paleomagnetic data. E-mail address: [email protected]. 0012-821X/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.epsl.2005.01.027 248 L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 There appears to be no compelling reason at this time to abandon the geocentric dipole hypothesis, which has provided such an excellent working model for so long. D 2005 Elsevier B.V. All rights reserved. Keywords: geomagnetic field; axial geocentric dipole hypothesis; sedimentary inclination error; paleosecular variation; Asian inclination anomaly 1. Introduction model, but with paleolatitudes predicted from geo- detic and global paleomagnetic data sets. The idea that the Earth’s magnetic field is well approximated by a geocentric axial dipole (GAD) is a very old one. It is central to much of modern 2. The birth of the geocentric axial dipole paleomagnetism which relies on records of the hypothesis geomagnetic field imprinted in rocks. The GAD hypothesis applied to paleomagnetic data provided Prior to his appointment as physician to Queen the first geophysical proof of continental drift and is Elisabeth I, William Gilbert (1544?–1603) inves- still the best way to reconstruct continents with tigated the magnetic properties of spherical speci- respect to paleolatitude and orientation relative to the mens of lodestone he called bterrellae,Q or blittle north pole. Nonetheless, despite centuries of study, Earths.Q He found that iron spikes aligned the limits of the GAD hypothesis are not precisely themselves on the terrellae in unexpected ways known. How much time is required to average the depending on the positions relative to the mag- field to that of a centered dipole? How much netic poles of the terrellae (see Fig. 1a). At the deviation from GAD can be expected from the equator, the spikes were aligned tangent to the time-averaged geomagnetic field? Has the field sphere. As they approached the poles, bthe more always been essentially dipolar, or was it more they are raised up by their versatory nature,Q and complex earlier in Earth’s history? Are discrepancies at the poles, the spikes pointed directly to the between geological and paleomagnetic predictions center of the sphere. the result of bbadQ recording of the magnetic field, Gilbert was not the first to consider the unrecognized crustal deformation, or strongly non- magnetic properties of the Earth and of rocks dipolar ancient magnetic fields? These are old (see [1]), but he seems to have been the first to questions, but have been the subject of much recent make a systematic study. In a great leap of insight effort. (perhaps aided by prior work of Petrus Peregrinus This paper is not a comprehensive review of the in the 13th century), he realized that the behavior history of geomagnetism. For that the reader is of his terrellae was similar to that of the Earth directed to a marvelous paper by Stern [1]. Nor is itself. He used a simple instrument for measuring this a thorough treatment of secular variation or the dip of the Earth’s magnetic field and showed time-averaged field models. We will begin with just how the dip could be transformed into latitude a brief tour of these subjects. We will consider how using a complicated graphical approach. He sedimentary inclination flattening affects the record- exuberantly proclaimed, bWe may see how far ing of the geomagnetic field, then discuss recent from unproductive magnetic philosophy is, how efforts at finding simple detection and correction agreeable, how helpful, how divine! Sailors when methods, in particular the elongation/inclination tossed about on the waves with continuous cloudy method of Tauxe and Kent [2]. Finally, we will weather, and unable by means of the coelestial consider a few case studies where the elongation/ luminaries to learn anything about the place of inclination method brings paleomagnetic data into the region in which they are, with a very slight agreement not only with a simple statistical field effort and with a small instrument are comforted, L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 249 Fig. 1. (a) Reproduction Gem Gilbert’s de Magnete [3] description of experiments with iron spikes near a globe made of lodestone. The Orbis Virtutis is the region within which the spikes responded to the globe. (b) Lines of flux for a dipole with moment m as a function of radial distance r and angle away from the pole h or equator k. At any point the angle that the field lines make with the local horizontal (tangent to the heavy circle) is the inclination I. (c) Definition of a virtual geomagnetic pole (VGP) as the piercing point of the centered dipole that would give rise to the direction (dashed line) at a given observation site P. and learn the latitude of the place.Q [3]. This was had a rough idea from sailors’ measurements of what the first statement of what has come to be known the deviation from true north was over much of the as the geocentric axial dipole (GAD) hypothesis. Earth. However, he mistakenly assumed that the Gilbert knew little of the physics of magnetism degree of variation was constant in time at a given (let alone trigonometry). Physics tells us that in the special case away from currents and changing electric fields, a magnetic moment m creates a Table 1 magnetic field H which is the gradient of a scalar Table of acronyms and terms potential field Vm (i.e., H=ÀjVm). Vm is a function Symbol Term and definitions of radial distance r and the angle away from the pole GAD Geocentric axial dipole h by (Table 1): m Magnetic moment H Magnetic field md r mcosh Vm Magnetic potential h, k, /, r Co-latitude, latitude, longitude, radius from V m ¼ 3 ¼ 2 : ð1Þ 4pr 4pr center of the Earth D, I Declination, inclination From the potential equation, it is possible to VGP Virtual geomagnetic pole m b Q calculate the field lines produced by the magnetic Pl Harmonic functions of cosh dipole m as shown in Fig. 1b. If we imagine the outer CP88/TK03 Statistical paleosecular variation model of [24] and [2] circle to be the surface of the Earth, the dip of the field b Q V Principal components of a set of directions. lines relative to horizontal ( inclination, I) varies V1 bAverage directionQ, direction along which progressively with latitude k from horizontal at the the data are concentrated. equator to vertical at the pole. We can replace the V3 Direction orthogonal to the average in which intricate graphical approach of Gilbert with the simple the data are least concentrated. V Direction orthogonal to V , V . trigonometric function known as the bDipole For- 2 1 3 E Ratio of variance along V2, V3. mulaQ: tanI=2 tank. f Flattening factor in the inclination error Gilbert knew that the Earth’s magnetic field was formula; ratio of tangents of not that of a perfect bar magnet. In such a field, inclinations of the ambient field versus that compasses would always point to the same pole. He recorded by the sediments. 250 L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 place and even proposed that variation could be used 3.

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