Theoretical Geomagnetism. Lecture 2: Self- Exciting Dynamos: Kinematic Theory

Theoretical Geomagnetism Lecture 2: Self- Exciting Dynamos: Kinematic Theory 1

2.0 What is a self-exciting dynamo? Dynamo = A device that converts kinetic energy into electromagnetic energy. Dynamos use motion of an electrical conductor through a magnetic field to induce electrical currents. These electrical currents also generate (secondary) induced magnetic fields that are sustained as long as energy is supplied. Self-exciting dynamo = No external fields or currents are needed to sustain the dynamo, aside from a weak seed magnetic field to get started. i.e. self-exciting dynamo is self-sustaining 2

2.0 What is a self-exciting dynamo? To generate and sustain magnetic fields self-exciting dynamos require: (i) A weak initial seed magnetic field (which can be removed later). (ii) A suitable arrangement of the motions of the electrical conductor, electric current pathways and resulting magnetic fields. (iii) A continuous supply of energy to drive the electrical conductor sufficiently fast for self-excitation to be possible. Many natural magnetic fields are thought to be produced by a self-exciting dynamo mechanism. e.g. Galactic magnetic fields, solar magnetic field, geomagnetic field. 3

Lecture 2: Self- exciting dynamos 2.1 Simple example 1: The disk dynamo 2.2 The induction equation and magnetic energy evolution 2.3 Simple example 2: Stretch-twist-fold dynamo 2.4 The Alpha-effect and the Omega-effect 2.5 Mean field dynamos 2.6 3D spherical dynamos 2.7 Summary 6

2.1 Simple example 1: The disk dynamo (a) u B 0 B 0 ω A N u D E + +++ + + ++ ++ +++ P N A B 0 D R (From Roberts, 2007) Disk is rotated in an applied magnetic field causing an induced EMF, but no closed circuit for current to flow (yet). 7

2.1 Simple example 1: The disk dynamo (b) W I I S 1 I S 2 ω B 0 A I u D R A (From Roberts, 2007) Now let a conducting wire join A to A, being maintained by sliding contacts ( brushes ). A current now flows and can be used to light the bulb, but the magnetic field due to current I doesn t reinforce B0. 8

2.1 Simple example 1: The disk dynamo ) B S 2 ω A B I S 1 W u J B B J D I W I B A B (From Roberts, 2007) Instead, let wire W loop round the disk. The induced field B produced by I now reinforces B0 in the plane of the disk. If rotation rate exceeds some critical value B0 can be removed => self-excited dynamo. 9

2.2.1 The magnetic induction equation Recall that in a moving, electrically conducting medium Ohm s law has a extra term: J = σ(e + u B) (1) i.e. motion across a magnetic field can generate a perpendicular current. Combining this with the Maxwell equations describing no magnetic monopoles, EM induction and Ampere s law yields: B t = (u B)+η 2 B (2) So, field changes can be produced by induction due to fluid motions or by Ohmic dissipation (magnetic diffusion) due to finite conductivity. 11

2.2.2 Magnetic energy evolution B/µ 0 Taking the dot product of with (2) and integrating over the over the conducting volume V we obtain the following terms: de m dt = V B B µ 0 t dv = V t ( B 2 ) dv 2µ 0 D J = V B µ (η 2 B) dv = 1 σ 0 V ( B) 2 + [B ( B)] dv = V J 2 σ dv 0 0 S (J B) σ ds W u = V B µ 0 [ (u B)] dv = 1 µ 0 V B [(B )u (u )B] dv = 1 µ 0 V B (B )u dv 1 2µ 0 S B 2 u ds Surface terms can be neglected if surface is rigid (no flow across it) and there is no electrical currents flowing at the boundary. 12

2.2.2 Magnetic energy evolution Taking the dot product of with (2) and integrating over the over the conducting volume V we obtain: de m dt = W u = D J = V V V B B µ 0 t dv = B µ 0 [ (u B)] dv = 1 µ 0 V B0 0 µ (η 2 B) dv = 1 σ V t ( B 2 ) dv 2µ 0 V B/µ 0 B [(B )u (u )B] dv = 1 µ 0 ( B) 2 + [B ( B)] dv = V 0 B (B )u dv 1 2µ V J 2 σ dv S S B 2 u ds (J B) ds σ 0 de m dt = W u + D J (3) Change in magnetic energy = Work done by flow + on magnetic field Ohmic Dissipation (always -ive) So, for the magnetic energy to grow, enough work must be done on the field by the fluid motions to overcome Ohmic dissipation. 13

2.3 Simple example 2: Stretch-twist-fold dynamo Special arrangements of moving solid conductors and wires can produce a self-exciting dynamo. But what type of fluid motions can do the same? Consider Stretch-Twist-Fold thought experiment dynamo of Vainshein and Zeldovich (1975): (a) Flow stretches flux tube to double its length. (b) Flow twists tube into a figure-8. (c) Magnetic diffusion acts where there are large field gradients. ((d) Flow folds loops on top of one another to form 2 linked loops the same size as original. (e) Left with 2 flux tubes very similar to the original... process can now repeat... Davidson (2011) 15

2.4.1 Toroidal and Poloidal fields Toroidal field (or flows) stay entirely on spherical surfaces: Poloidal field (or flows) cut across spherical surfaces: Axisymmertic examples of Toroidal and Poloidal fields (Credit: Graeme Sarson) 17

2.4.1 Toroidal and Poloidal fields (An aside) Previously, we represented the (geo)magnetic field in an insulating medium (mantle) as B = rv. Any magnetic field in an arbitrary medium can also be represented as the sum of a toroidal and a poloidal field: B = B T + B P = r T (r,, )r + r r P (r,, )r This decomposition holds for any solenoidal vector field, e.g. the current density: J = J T + J P. The curl operation maps toroidal fields on poloidal ones and vice versa, i.e. toroidal currents generate poloidal magnetic fields and vice versa. 18

2.4.2 The Omega-effect (a) (b) (From Roberts, 2007) Conversion of poloidal field to toroidal field. Caused by differential rotation (shear) of toroidal flows. 19

2.4.3 The Alpha-effect (a) u Regeneration of poloidal field from toroidal field. (can also sometimes be used to produce toroidal field from poloidal field) Caused by upwelling poloidal flows possessing vorticity. (b) (c) R ω B B j B (From Roberts, 2007) 20

2.4.3 Simple example 3: Alpha-Omega dynamo Omega-effect: Poloidal field -> Toroidal field (From Love, 1999) Alpha-effect: Toroidal field -> Poloidal field 21

2.5.1 Mean-field electrodynamics We begin with the magnetic induction equation: B t = (u B)+η 2 B (2) We separate the velocity and magnetic field into mean and fluctuating parts: u(r,t)=u 0 (r,t)+u (r,t), with =0 B(r,t)=B 0 (r,t)+b (r,t), with =0 < B(r,t) >= B 0 (r,t) and < u(r,t) >= U 0 (r,t) This separation is valid if the fields are characterized, for example, by 2 length scales: a large scale L and a small scale l0 and the averaging is carried out over an intermediate lengthscale a where l0 <<a<< L such that: <ψ(r,t) >= 3 4πa 3 ψ(r + ξ,t) d 3 ξ (5) ξ <a Small scale fluctuations are assumed random so they average to zero. 23 (4)

2.5.1 Mean-field electrodynamics Substituting from (4) into (2) and carrying out the averaging operation on each term we obtain the the mean field induction equation: B 0 t = (U 0 B 0 )+ E + η 2 B 0 (6) E = where is a non-zero mean EMF arising from the product of random, small scale fluctuations of the field and flow. Subtracting (6) from (4) we obtain an induction equation for the evolution of the fluctuating part of the magnetic field: b t = (U 0 b )+ (u B 0 )+ G + η 2 b (7) where G = u b ( Pain in the neck term!) 24

2.5.1 Mean-field electrodynamics In order to make progress in solving (6) we need to find a way to express E in terms of the mean fields U 0 and B 0. We proceed assuming that fluctuations in the magnetic field are driven solely by the large scale field. α ij β ijk B 0 Under this assumption we can further assume that and are linearly related, so E and B 0E will also be linearly related so we can postulate an expansion for of the form: E i = α ij B 0j + β ijk B 0j x k +... and are pseudo-tensors determined by the statistics of the small scale motions (turbulence). In the simplest isotropic case: α ij = αδ ij and β ijk = βϵ ijk and so E = αb 0 β( B 0 )+... (8) b b B 0 25

2.5.1 Mean-field electrodynamics Keeping only the 2 terms given on the RHS of (8) and substituting back into (6) gives B 0 t = [(U 0 B 0 )+αb 0 ]+(η + β) 2 B 0 (9) This is the mean field induction equation for isotropic turbulence. α is now seen to be a regenerative term which can help the large scale magnetic field to grow. β is seen to be a turbulent diffusivity But, what mathematical form does take and what does it physically represent? This is a complicated and controversial issue. α See the appendix for one answer in a very simple scenario... 26

2.5.2 Simple example 4: alpha-effect dynamo Let us consider a simple mean field dynamo that arises due to the term. If =0 and defining an effective diffusivity (9) can be written as α U 0 η e = η + β B 0 t = α( B 0 )+η e 2 B 0 We seek solutions that have the special Beltrami property B 0 = KB 0 Consequently, 2 B 0 = ( B 0 )= K 2 B 0 So the induction equation simplifies to: B 0 t =(αk η e K 2 )B 0 27

2.5.2 Simple example 4: alpha-effect dynamo From the induction eqn: B 0 Separating the variables: t 1 B 0 db 0 = =(αk η e K 2 )B 0 (αk η e K 2 )dt Integrating: i.e. Thus, if ln B 0 (r,t) ln B 0 (r, 0) = ( αk η e K 2 ) t B 0 (r,t)=b 0 (r, 0)e (αk η ek 2 )t α>η e K => Mean field will grow exponentially and we have a self-exciting dynamo!! Since K L 1 this will happen if the length scale of the mean field is sufficiently large. 28

2.5.3 Shortcomings of mean field dynamo theory Mean field dynamo theory is very useful to help develop intuition concerning how self-exciting dynamos can come about and allows analytic/simple numerical solutions to be found. However, it involves many major assumptions and can t be rigorously justified when applied to geophysical and astrophysical systems. Major weaknesses include: (1) Difficulty in deriving form of and for realistic scenarios. Instead,these are often picked in an ad-hoc manner to fit observations. α β α (2) Theory of how and are modified when magnetic field become strong enough to affect flows is unclear. To accurately model the geodynamo, we must leave behind mean field theory and consider numerical solution of the full induction eqn. 29 β

2.6.1 Kinematic dynamo theory: Solution as an eigenvalue problem Reminder: We are considering the KINEMATIC dynamo problem. i.e. Assume the flow is known and ignore any back-reaction of the generated field on the flow Once again we start from the induction eqn: B t = (u B)+η 2 B We then seek normal mode solutions of the form: B(r,t)=Re[ B(r)e pt ] This transforms the induction eqn into a linear eigenvalue problem: p B = (u B)+η 2 B 31

2.6.1 Kinematic dynamo theory: Solution as an eigenvalue problem The solutions of the Linear eqn: p B = (u B)+η 2 B Define a sequence of complex eigenvalues that can be ordered: Re[p 1] Re[p 2 ] Re[p 3 ] Re[p 4 ]... Re[p 1 ] > 0 B(r,t)=Re[ B 1 (r)e p 1t ] If then the corresponding field will exhibit exponential growth, which is oscillatory if Im[p 1 ] 0. Challenge: To solve this eigenvalue problem given any 3D flow. 32

2.6.2 Numerical solutions for 3D flows Requires numerical solution, for example, by expanding the field and flows into of toroidal and poloidal parts, each represented as a sum of vector spherical harmonics: u = i t m l + s m l where t m l and s m l are the toroidal and poloidal components t m l = ˆrt m l (r, t)y m l (θ, φ), s m l = ˆrs m l (r, t)y m l (θ, φ) Gubbins and co-workers have carried out extensive tests of the steady flows: u = ϵ 0 t 0 1 + ϵ 1 s 0 2 + ϵ 2 s 2c 2 + ϵ 3 s 2s 2 Differential Rotation Meridional Flow Spiralling Convection Cells Flow Streamlines: Meridional Section Meridional Section Equatorial Section 33

2.6.3 Results of parameter studies M= Fraction of flow Energy in meridional circulation. D= Fraction of flow Energy in differential rotation. Da Q a D e (Axial dipole for lowest Rm) (Axial quadrupole for lowest Rm) (Equatorial dipole for lowest Rm) White lines: oscillatory solns (From Gubbins, 2008) In many parts of parameter space NO DYNAMOS are found! In regions where solutions with axial/equatorial dipole symmetry are found, smaller embedded regions are found where oscillatory solutions can occur. 34

2.7 Summary: self-assessment questions (1) How does a self-exciting disk dynamo work? (2) Can you describe the alpha effect and the omega effect? (3) Can you derive the mean field dynamo equations and solve them for simple cases? (4) How does one determine numerically whether a 3D flow will acts as a self-exciting dynamo? Note: Everything in this lecture ignored the influence of the magnetic field on the fluid motions... not fully self-consistent! Next time: Core dynamics: Rotation, Convection and the Lorentz force. 36

References - Davidson, P. A., (2001) An introduction to magnetohydrodynamics, Cambridge University Press. - Gubbins, D., (2008) Implications of kinematic dynamo studies of the geodynamo. Geophysical Journal International, Vol 173, 79-91. - Love, J. J, (1999) Reversals and excursions of the geodynamo. Astronomy and Geophysics, Vol 40, 6, 14-19. - Moffatt, H.K., (1978) Magnetic field generation in electrically conducting fluids, Cambridge University Press. - Roberts, P.H., (2007) Theory of the geodynamo. In Treatise on Geophysics, Vol 8 Geomagnetism, Ed. P. Olson, Chapter 8.03, pp.67-102. (especially section 8.03.2). 37

Appendix A: The significance of alpha α To illustrate the significance of we shall derive its form for a simple case when the small scale flow takes the form of a circularly polarized wave travelling in the z direction: u = u 0(sin(kz ωt), cos(kz ωt), 0) (10) B 0 We consider that is constant and // to z. This is possible since is independent of. Further, we assume = 0. Then (7) is: b t =(B 0 )u + [u b ]+η 2 b (11) B 0 U 0 α One can show (11) is satisfied with flow (10) provided b = B 0u 0k ω 2 + η 2 k 4 (ηk2 cos γ ω sin γ, ω cos γ ηk 2 sin γ,0) with γ =(kz ωt) Therefore, u b = B 0u 2 0 k 3 η ω 2 + η 2 (0, 0, 1) k4 and E == u 2 0 k 3 η ω 2 + η 2 k 4 B 0 so that α = u 2 0 k 3 η ω 2 + η 2 k 4 38

Appendix A: The significance of alpha To interpret this result, we note first that the helicity density flow is Therefore α H = u ( u )=ku 2 0 is directly proportional to the helicity density: α = k2 η ω 2 + η 2 k 4 H H of the u u So, in this example, when is a circularly polarized wave then is also a circularly polarised wave, but is phase shifted relative to. u b B 0 The resulting is uniform and parallel to. It is the phase shift between u and b due to the diffusivity η which leads to the non-zero value of α. Although this derivation is rather special, similar results hold for a random superposition of such waves. It is however very difficult to derive the form of in general and its form is often chosen in an ad-hoc manner to match observations. α 39 b

Appendix B: Axisymmetric Mean Field Dynamos Several simple and physically interesting mean field dynamos arise in the case when the mean field is axisymmetric. In this case it is convenient to work in cylindrical polar co-ords (z, s, φ) and write: B 0 (s, z) =B(s, z) φ + B p (s, z) where B p = (A(s, z) φ) U 0 (s, z) =U(s, z) φ + U p (s, z) In this co-ordinate system the mean field induction equation for the evolution of the B component in the φ ( ) ( ) direction is: B B U t = s(u p ) + s(b p ) +( E) φ + η ( 2 1s ) s s 2 B B p = (A(s, z) φ) The remaining component associated with can be uncurled to yield an equation for the evolution of A: A t = s(u p ) ( A s ) + E φ + η ( 2 1s ) 2 A where as before we assume: E == αb 0 40

Appendix B: Axisymmetric Mean Field Dynamos B t = s(u p ) ( B s ) A t = s(u p ) + s(b p ) ( A s ) ( U s ) + E φ + η +( E) φ + η ( 2 1s ) 2 B ( 2 1s ) 2 A Terms associated with alpha-effect Terms associated with omega-effect To achieve self-exciting dynamo action must have continuous cycle of: (i) Magnetic Energy transferred from B p = (A(s, z) φ) to B φ. (ii) Magnetic Energy transferred from B φ to B p = (A(s, z) φ). The most important terms in transferring energy can be used to classify the form of dynamo action: E φ ( E) φ s(b p )(U/s) α 2 Yes Yes No αω Yes No Yes Yes Yes Yes 41