1. GEOMAGNETISM: a dynamo at the centre of the Earth Lecture 2 How the dynamo works

2. OVERVIEW Preliminaries: timescales, waves, instabilities, symmetry Non-magnetic convection Magnetoconvection Kinematic dynamo action Nonlinear, self-consistent dynamos

3. TIMESCALES Acoustic (seismic) waves1 min Gravity waves1 hr Inertial waves1 day Alfven waves10 yr Slow (MAC) waves100 yr Overturn time1 kyr Dynamo waves10 kyr Magnetic diffusion time15 kyr Thermal, viscous diffusion time100 Gyr

4. SYMMETRY Rotation has cylindrical symmetry The core has spherical symmetry These combine to give symmetry under inversion through the origin Changing the sign of the magnetic field does not alter the governing equations… …nor does translation in time

5. EQUATIONS ARE NOT CHANGED BY REFLECTION IN THE EQUATOR (E)

6. POSSIBLE SYMMETRIES Reflection through the equatorE S …with field reversalE A Field reversalI Rotation about polar axisP S …with field reversalP A Inversion through originO S …with field reversalO A

7. GROUP TABLE FOR 180 o POLAR ROTATION P 2

8. GI Taylor’s experiment to verify Proudman’s theorem: “fluid flow does not vary along the rotation axis”

9. BUSSE ROLLS

10. NON-DIMENSIONAL PARAMETERS Rayleigh number Ekman number Prandtl number

11. Limit as Rayleigh number Roll wavenumber Drift rate ASYMPTOTIC FORMULAE

12. SCALING TO THE EARTH Ekman number E=10 -15 ~ 10 -9 Critical Rayleigh number R a c =1.63 10 12 Number of rolls 1000 Drift rate -0.25 10 6 (in viscous diffusion times)

13. MAGNETOCONVECTION Elsasser number Large scale Lower Rayleigh number Positive drift rate

14. CONVECTION ROLLS Non-magnetic, E=10 -5 Magnetoconvection, Elsasser number =1

16. KINEMATIC DYNAMOS Specify a fluid flow Solve the induction equation for magnetic field Test for exponential growth or decay Magnetic Reynolds number measures the flow strength: Steady flow gives steady or oscillatory fields at a critical

17. REQUIREMENTS FOR DYNAMO ACTION Nonaxisymmetric field (Cowling) Radial motion Sufficiently large R m Sufficiently complicated flow (helicity)

18. COWLING’S LAST THEOREM? Nothing simple works… …and even when it does… proving that it works… is not as simple as it should be

19. DIFFERENTIAL ROTATION (omega effect)

20. HELICITY (alpha effect)

21. STRETCH-TWIST-FOLD

22. MEAN FIELD THEORY Small scale flow replaced by alpha effect (Braginsky) non-axisymmetric flow replaced by alpha effect Remember contribution to diffusivity (Braginsky) and contribution to large scale flow (effective meridian circulation) Solve axisymmetric equations

23. KUMAR-ROBERTS FLOW Meridian circulation (M) 2 convection rolls Differential rotation (D)

24. FIELD SYMMETRIES DaDeQa Axial dipole Equatorial dipole Axial Quadrupole

25. Kumar-Roberts Kinematic Dynamo D (westward) M (poleward)

26. D=0.95,M=0.00 “Braginsky”

27. D=0.10,M=0.30

28. D=-0.95,M=0.0 “Braginsky” quadrupole

29. NONLINEAR DYNAMO Momentum equation: rate of change of flow= inertia+coriolis+pressure+buoyancy+ viscosity+magnetic Heat equation: rate of change of temperature= advection+diffusion Induction equation: rate of change of magnetic field= advection+stretching+diffusion

30. NONLINEAR EFFECTS Magnetic field reaches a maximum value Time dependence can be more complex… …including reversals There is no longer the freedom to choose the flow The flow may resemble magnetoconvection, but there may be behaviour specific to the type of magnetic field generated

31. JB TAYLOR’S CONDITION Azimuthal magnetic torques on all cylindrical surfaces with axes parallel to the rotation axis must be zero…...or rapid oscillations develop that rapidly re- establish the torque as zero

32. WEAK FIELD REGIME Small Elsasser number A dynamo developed buy Busse (1975) Magnetic fields exert only a small force Convection looks like non-magnetic convection Magnetic fields generated by helicity from convection rolls + flow along the rolls induced by the boundary

33. STRONG FIELD REGIME Elsasser number about 1 Magnetic torques balance Coriolis torques JB Taylor condition satisfied Convection scales like magnetoconvection

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

36. 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

37. 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

38. 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?