On Modeling Magnetic Fields on a Sphere with DipolesA and Quadrupoles<*Nfc/ JL GEOLOGICAL SURVEY PROFESSIONAL PAPER 1118 On Modeling Magnetic Fields on a Sphere with Dipoles and Quadrupoles By DAVID G. KNAPP GEOLOGICAL SURVEY PROFESSIONAL PAPER 1118 A study of the global geomagnetic field with special emphasis on quadrupoles UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON: 1980 UNITED STATES DEPARTMENT OF THE INTERIOR CECIL D. ANDRUS, Secretary GEOLOGICAL SURVEY H. William Menard, Director Library of Congress Cataloging in Publication Data Knapp, David Goodwin, 1907- On modeling magnetic fields on a sphere with dipoles and quadrupoles. (Geological Survey Professional Paper 1118) Bibliography: p. 35 1. Magnetism, Terrestrial Mathematical models. 2. Quadrupoles. I. Title. II. Series: United States Geological Survey Professional Paper 1118. QC827.K58 538'.?'ol84 79-4046 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 Stock Number 024-001-03262-7 CONTENTS Page Page /VOStrHCt ™™-™™™™"™-™™™"™«™™-«—"—————————————— j_ Relations with spherical harmonic analysis—Continued Introduction——————————-—--——-———........—..——....... i Other remarks concerning eccentric dipoles ———— 14 Characteristics of the field of a dipole —————————————— 1 Application to extant models -—————————-——- Multipoles ——————————-—--——--—-——--——-——...._. 2 14 E ccentric-dipole behavior ———————————————• The general quadrupole ——————-———._....................—.— 2 20 Results of calculations --—--—------——-——-—- 21 Some characteristics of quadrupole fields -————————- 4 Additional results ———————————————— Character of lines of force ———————————————— 6 26 Interconversion of quadrupoles —————————————— $ Hypothesis on the character of the quadrupole —————- 27 Relations with spherical harmonic analysis —————————— 9 Some further implications of the quadrupole constraint — 29 Quadrupole secular change and its conservative aspect — 12 Some qualitative remarks on octupoles ——————.—... 34 The eccentric dipole ——————————————————— 13 Summary and conclusions -----———————————-—~ 35 Fertile and sterile parts of the quadrupole ———————— 13 Selected references —————————————————————— 35 ILLUSTRATIONS Page FIGURES 1-4. Sketches of: 1. Linear quadrupoles -—--————————-———---——————--———————————...——————— 2 2. Three conceptions of the rhombic quadrupole ————-——--———————---—————————-----——— 3 3. Linear quadrupoles on a sphere directed toward P0 and Pj ——————--—-—-——-——-—---———--— 3 4. Attitude of quadrupole shown in figures 5-7 -----——————————————————-———-——-————-— 6 5-7. Graphs of normal-quadrupole field on the sphere showing: 5. Potential, vertical intensity, and inclination ————————————————————————......————— 7 6. Magnetic meridians, declination, and horizontal and total intensity ——————————————————-—-— 7 7. North and east components ———--—--------———--—--—-_-———————————-———————————————— 7 8-11. Sketches of: 8. Dipole field line -.-----.-------------------------------------.---------.-----.----------------------------------------------------------------------------———- 8 9. Linear-quadrupole field line ---------—————-———————-—-——————————--——--—-----——- 8 10. Normal-quadrupole field line-----------------------------—--—--—-———-—-—-----————————————— 8 11. Relations of cardinal axes to primitive lines and identification of angular parameters ————---————— 11 12-17. Graphs of drift of the geomagnetic quadrupole: 12. Primary axis since 1550 —————————————————————————————————————————————— 15 13. Primary axis since 1930 ---------------------------------———-——-——---————————-———...——— jg J-^T.1 A. J.1/CUT?pH \^CLL(* ai*Hinfi U111CIJ. 1 CUVlOavic OlllVsCGITI/"»O J.tltlV/1 ^^O -.—..—_.._-.___— —————————_ _..______ _ _....._..—————————_—.._———..—————.——— j_1 7i 15. Red cardinal axis since 1930 ————————————————————————————————————————————— 17 J.O. XJlUc CHiQlrlHI HX1S 31HC6 J.O«5vJ ———————---------.————-—---——----————-———————————--—--—————————————————---»«-•••-•--•-- J.o j.1 7i • JJAUCRind \~>CLLi^suvlifisil miicu. d.A.i.0avio onicccini^o J.1 Q^flzftj\j ____. _________ __ .——————— _ ....——__ __ __ ..._———_———————.—————._.———..———————————.———— j.1Q ^ 18-20. Graphs of change of U: 18 Since 1550 _________________________________________________________ 20 19 Since 1925 ______________________________________—_________________ 21 £A\J9fl f O111CC7C!inr»f> 1J. Q117-1 fl\J _ ———————"—-«———"-———---———--————————-———™...™......™....™.....™....———..................................... —.... ————— 99£t£t 21. Direct plot of change of the quadrupole moment ——————————————————————————————————————— 23 22. Sheared plot of change of the quadrupole moment.----.--.-——-————-————————-—-—--—————---—--— 24 23. Graph of coordinates of displacement vector for eccentric dipole —————-—-——--————--——--————— 27 24. Graph of change of dc for selected analyses ———-----------------———————————-——------———-———————-—————- 28 25. Graph of coordinates of the centered-dipole axis ————--——-—————----——-——————————————-—————— 30 26. Graph of quadrupole rotation rate using Class II analyses —————-—--——-—-——————----—-——--—----- 31 27. Graph of center of clockwise quadrupole rotation using chiefly Class I analyses ———————————————————— 32 28. Graph of center of clockwise quadrupole rotation using chiefly Class II analyses —-——-——-—-——--—--——— 33 29. Schematic diagram of current loops in normal quadrupole —--———-—-----————---—-—------------—— 33 30. Sketch of arrangement of four dipoles forming a cubic octupole, and equivalent quatrefoil array of coplanar current lOOyS ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------^in o4 IV CONTENTS TABLES Page TABLE 1. Expressions for the fields of dipoles and quadrupoles ——————————————————————— — ————————————— — . 5 «. '9fU3.Q.rupO16 |j£Lr£lIll6u6rS OI SpGCiriC IlGlQ IT1OCL61S --------------------------------------------------------------------------- —-—-——"—--— ™™-—™™—-™™™- J_Q Q fjroiirjiriff of pxttirit snh.6riC3.1 hflrmoriic flmilvsps -•-------•-•-»•--•-------------------"----—-----•"--—-----—--«----—"------—----------—--—--——•—»-»• 2^S 4 Ecc6ntriC"diriol6 nt 5. Changes of coefficients caused by imposing the constraint for U = 1 — ----- — - — - — — — - — - — — —— — - — — — — —— -- — - 34 ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES By DAVID G. KNAPP model in fidelity to nature but giving a somewhat truer ABSTRACT picture than the centered dipole alone), but also for its This paper assists in the understanding of the global geomagnetic bearing upon the characteristics of models comprising field as it is manifested in models slightly more complex than the an array of two or more radial or other eccentric di- centered dipole, with special emphasis on the quadrupole, which poles. among the harmonic components is the chief determinant of the The centered dipole fails to embody two striking nondipole field configuration. To this end it examines the geometric properties of the three kinds of quadrupoles, as well as their inter- features of the world charts, namely (1) the oblique or convertibility, and the ways in which they can be combined or re­ "corkscrew" aspect of the agonic lines, and (2) the solved into constituent parts. Improved methods are developed for elongation of the intensity loops enclosing the Arctic establishing the quadrupole parameters (especially the cardinal dip pole. The eccentric-dipole model does depict feature axes) from spherical harmonic coefficients and for using the various (1), which, however, is not intrinsic to the field but extant analyses to examine the secular change of the quadrupole. Earlier reports of the quadrupole's westward drift are confirmed, its rather an effect of its relation to the coordinate system rotation being found to be about 15 minutes per year, but the center (reflecting a longitude difference between the dipole's of clockwise rotation is markedly displaced from the geographic displacement vector and the meridian plane of the cen­ pole and is situated in the region of the Aleutian Islands. A graphic tered dipole). To account for (2), a genuine field charac­ display clarifies the disparities among different models, promotes teristic, requires still greater complexity in .the model. the study of relative benefits of refinements in spherical harmonic analysis, and points the way toward a definitive assessment of the quadrupole and especially of its secular change. The way in which CHARACTERISTICS OF THE FIELD OF A DIPOLE one part of the quadrupole combines with the centered dipole to pro­ duce the eccentric dipole is also examined (with some possible bear­ The simplest useful model is the one developed by ing on radial-dipole models). Support is presented for the hypothesis Biot (Humboldt and Biot, 1804), namely, a magnetic that the geomagnetic quadrupole tends to hold rather closely to the aspect of a "normal" quadrupole—one with identical configurations dipole at the center of the globe. The basic notion of a for its positive and negative regions. Some properties of octupoles dipole grew out of the then rather novel concept of are discussed qualitatively. magnetic point poles (or as Biot preferred to say, "cen­ ters of action") and represents the limiting case of a INTRODUCTION pair of point poles of opposite kind, as they are caused to approach one another to within an infinitesimal sep­ Ever since the time of William Gilbert nearly four aration, while their