1. GEOMAGNETISM: a dynamo at the centre of the Earth Lecture 3 Interpreting the Observations

2. OVERVIEW Historical and paleomagnetic data Historical record gives good spatial resolution Paleomagnetic record covers a long time interval Measurements are interpreted in terms of a core field generated by a dynamo Field morphology interpreted in terms of the dynamics Secular variation in terms of core flows Is the geodynamo unstable?

3. HISTORICAL RECORD Accuracy limited by crustal field mainly Global coverage => good spatial resolution Many components measured Known dates Short duration: 1500-2000AD (500 years)

4. Length of day measured and predicted (Jackson et al. 1993)

5. ~1500Start of navigational records 1586Robert Norman measures inclination in London 1600William Gilbert publishes de Magnete 16?? Jacques l’Hermite’s voyage across Pacific 1695Edmund Halley measures D in Atlantic 1715Feuille measures I in Atlantic and Pacific 1777James Cook’s voyages; solves longitude problem 1839Gauss measures absolute intensity 1840Gottingen Union Observatories set up 1840Royal Navy exploration of Southern Oceans 1840 James Ross expedition to Antarctica 185?Suez Canal built 1887Challenger expedition 1900--First magnetic surveys, first permanent observatories 1926Carnegie burns out in Apia harbour 1955Proton magnetometer starts widespread aero- and marine surveys 1966First total intensity satellites (POGO) 1980Magsat 2001--Oersted, Champ, etc: decade of magnetic

9. Voyage of HMS Challenger

10. Halley and the “Paramour”

11. Historical data (after Jackson, Walker & Jonkers 2000)

12. POTENTIAL THEORY: uniqueness requires measurements on the boundary of: The potential V Normal derivative of the potential =Z… …but not F (Backus ambiguity) North component X…... but not East component Y D and I (to within a multiplicative constant) D and H on a line joining the poles

13. SPHERICAL HARMONIC EXPANSIONS Differentiating the potential gives the magnetic field components Setting r=a the Earth’s radius gives a standard inverse problem for the geomagnetic coeficients in terms of surface measurements Setting r=c the Core radius gives the magnetic field on the core surface

14. DATA KERNELS (Gubbins and Roberts 1983) The magnetic potential V at radius r is an average over the whole core surface: where is the angular distance between and (r’, Then where This is the data kernel for the inverse problem of finding the vertical component of magnetic field at the core surface from measurements of vertical component of magnetic field at the Earth’s surface. The data kernel for a horizontal component measurement, N h, is found by differentiating with respect to

15. DATA KERNELS

16. Smoothing constraint Data plane

18. LEAST SQUARES L1 NORM (double exponential)

19. Declination AD1600 Declination AD1990

23. The Tangent Cylinder

25. PALEO/ARCHEOMAGNETIC DATA Locations limited 10x less accurate than direct measurement Rarely is the date known accurately Hence rarely more than one location at a time Record is very long duration (Gyr)

26. THE TIME-AVERAGED PALEOMAGNETIC FIELD LAST 5Ma

30. Hawaiian data last 50 kyr from borehole data and surface flows (Teanby 2001)

31. Critical Rayleigh number for magnetoconvection E=10 -9

32. AN IMPORTANT INSTABILITY? Nobody has yet found a dynamo working in a sphere in the limit (Fearn & Proctor, Braginsky, Barenghi, Jones, Hollerbach) Perhaps there is none because the limit is structurally unstable Small magnetic fields lead to small scale convection and a weak-field state, which then grows back into a strong-field state This may manifest itself in erratic geomagnetic field behaviour

33. Time scale to change B in outer core: 500 yr Time scale in inner core (diffusion) 5 kyr STABILISING THE GEODYNAMO

34. DYNAMO CATASTROPHE The Rayleigh number is fixed The critical Rayleigh number depends on field strength Vigour of convection varies with supercritical R a … So does the dynamo action If the magnetic field drops, so does the vigour of convection, so does the dynamo action The dynamo dies

35. NUMERICAL DIFFICULTIES At present we cannot go below The resulting convection is large scale The large E prevents collapse to small scales… …and therefore the weak field regime Hyperdiffusivity suggests smaller E…...but the relevant E for small scale flow is actually larger

36. CONCLUSIONS We are still some way from modelling the geodynamo, mainly because of small E The geodynamo may be unstable, explaining the frequent excursions, reversals, and fluctuations in intensity Is the geodynamo in a weak-field state during an excursion? If not, what stabilises the geodynamo?