Geomagnetic Dynamo Theory

Transcription

1 Max Planck Institute for Solar System Research Katlenburg-Lindau, Germany International School of Space Science Geomagnetism and Ionosphere L Aquila, Italy 7-11 April 2008

2 Outline Introduction 1 Introduction Need of a geodynamo Homopolar dynamo Geodynamo hypothesis 2 Pre-Maxwell theory Induction equation Alfven theorem Magnetic Reynolds number and flux expulsion Poloidal and toroidal magnetic fields 3 Antidynamo theorems Parker s helical convection Mean-field theory Mean-field coefficients Mean-field dynamos 4 Equations and parameters Proudman-Taylor theorem Convection in rotating sphere Taylor s constraint 5 A simple model Interpretation Advanced models Reversals Power requirement 6 Literature

3 Need of a geodynamo Need of a geodynamo Homopolar dynamo Geodynamo hypothesis Earth and many other celestial bodies possess a large-scale, often variable magnetic field Decay time τ = L 2 /η magnetic diffusivity η = c 2 /4πσ, electrical conductivity σ Earth τ yr Variability of geomagnetic field Dynamo: u B j B u Faraday Ampere Lorentz motion of an electrical conductor in an inducing magnetic field induction of electric current Self-excited dynamo: inducing magnetic field created by the electric current (Siemens 1867)

4 Homopolar dynamo Introduction Need of a geodynamo Homopolar dynamo Geodynamo hypothesis B u uxb electromotive force u B electric current through wire loop induced magnetic field reinforces applied magnetic field self-excitation if rotation Ω > 2πR/M is maintained where R resistance, M inductance

5 Geodynamo hypothesis Need of a geodynamo Homopolar dynamo Geodynamo hypothesis Larmor (1919): Magnetic field of Earth and Sun maintained by self-excited dynamo Homogeneous dynamo (no wires in Earth core) complex motion necessary Kinematic (u prescribed, linear) Dynamic (u determined by forces, including Lorentz force, non-linear)

6 Pre-Maxwell theory Introduction Pre-Maxwell theory Induction equation Alfven theorem Magnetic Reynolds number and flux expulsion Poloidal and toroidal magnetic fields Maxwell equations: c B = 4πj + E t cgs system, vacuum, B = H, D = E, c E = B t Basic assumptions of MHD:, B = 0, E = 4πλ u c: system stationary on light travel time, no em waves high electrical conductivity: E determined by B/ t, not by charges λ c E L B T E B 1 c L T u c 1, E plays minor role : e el e m E2 B 2 1 E/ t c B E/T cb/l E u B c u2 1, displacement current negligible c2 Pre-Maxwell equations: c B = 4πj, c E = B t, B = 0

7 Pre-Maxwell theory Introduction Pre-Maxwell theory Induction equation Alfven theorem Magnetic Reynolds number and flux expulsion Poloidal and toroidal magnetic fields Pre-Maxwell equations Galilei-covariant: E = E + 1 c u B, B = B, j = j Relation between j and E by Galilei-covariant Ohm s law: j = σe in resting frame of reference, σ electrical conductivity j = σ(e + 1 c u B) Magnetohydrokinematics: c B = 4πj c E = B t B = 0 j = σ(e + 1 c u B) Magnetohydrodynamics: additionally Equation of motion Equation of continuity Equation of state Energy equation

8 Induction equation Introduction Pre-Maxwell theory Induction equation Alfven theorem Magnetic Reynolds number and flux expulsion Poloidal and toroidal magnetic fields Evolution of magnetic field ( B j = c E = c t σ 1 ) ( c c u B = c 4πσ B 1 ) c u B ( ) c 2 = (u B) 4πσ B = (u B) η B with η = c2 = const magnetic diffusivity 4πσ induction, diffusion (u B) = B u + (B )u (u )B expansion/contraction, shear/stretching, advection

9 Alfven theorem Introduction Pre-Maxwell theory Induction equation Alfven theorem Magnetic Reynolds number and flux expulsion Poloidal and toroidal magnetic fields Ideal conductor η = 0 : B t = (u B) Magnetic flux through floating surface is constant : d B df = 0 dt F Proof: 0 = BdV = B df = B(t) df B(t) ds udt, F F C F B(t +dt) df { B(t) df = B(t +dt) B(t) } df B ds udt F F C ( ) ( ) B = dt t df B ds u = dt (u B) df B ds u C C ( ) = dt u B ds B ds u = 0 C C

10 Alfven theorem Introduction Pre-Maxwell theory Induction equation Alfven theorem Magnetic Reynolds number and flux expulsion Poloidal and toroidal magnetic fields Frozen-in field lines impression that magnetic field follows flow, but E = u B and E = c B/ t B t = (u B) = B u + (B )u (u )B (i) star contraction: B R 2, ρ R 3 B ρ 2/3 Sun white dwarf neutron star: ρ [g cm 3 ]: (ii) stretching of flux tube: Bd 2 = const, ld 2 = const B l (iii) shear, differential rotation

11 Differential rotation Introduction Pre-Maxwell theory Induction equation Alfven theorem Magnetic Reynolds number and flux expulsion Poloidal and toroidal magnetic fields B φ / t = r sin θ Ω B p

12 Magnetic Reynolds number Pre-Maxwell theory Induction equation Alfven theorem Magnetic Reynolds number and flux expulsion Poloidal and toroidal magnetic fields Dimensionless variables: length L, velocity u 0, time L/u 0 B t = (u B) R 1 m B with R m = u 0L η as combined parameter laboratorium: R m 1, cosmos: R m 1 induction for R m 1, diffusion for R m 1, e.g. for small L example: flux expulsion from closed velocity fields

13 Flux expulsion Introduction Pre-Maxwell theory Induction equation Alfven theorem Magnetic Reynolds number and flux expulsion Poloidal and toroidal magnetic fields (Weiss 1966)

14 Poloidal and toroidal magnetic fields Pre-Maxwell theory Induction equation Alfven theorem Magnetic Reynolds number and flux expulsion Poloidal and toroidal magnetic fields Spherical coordinates (r, ϑ, ϕ) Axisymmetric fields: / ϕ = 0 B(r, ϑ) = (B r, B ϑ, B ϕ ) B = 0 B = B p + B t poloidal and toroidal magnetic field B t = (0, 0, B ϕ ) satisfies B t = 0 B p = (B r, B ϑ, 0) = A with A = (0, 0, A ϕ ) satisfies B p = 0 B p = 1 ( r sin ϑaϕ, r sin ϑa ) ϕ, 0 r sin ϑ r ϑ r axisymmetric magnetic field determined by the two scalars: r sin ϑa ϕ and B ϕ

15 Poloidal and toroidal magnetic fields Pre-Maxwell theory Induction equation Alfven theorem Magnetic Reynolds number and flux expulsion Poloidal and toroidal magnetic fields Axisymmetric fields: j t = c 4π B p, j p = c 4π B t r sin ϑa ϕ = const : field lines of poloidal field in meridional plane field lines of B t are circles around symmetry axis Non-axisymmetric fields: B = B p + B t = (Pr) + (Tr) = (r P) r T r = (r, 0, 0), P(r, ϑ, ϕ) and T(r, ϑ, ϕ) defining scalars B = 0, j t = c 4π B p, j p = c 4π B t r B t = 0 field lines of the toroidal field lie on spheres, no r component B p has in general all three components

16 Cowling s theorem (Cowling 1934) Introduction Antidynamo theorems Parker s helical convection Mean-field theory Mean-field coefficients Mean-field dynamos Axisymmetric magnetic fields can not be maintained by a dynamo. Sketch of proof: electric currents as sources of the magnetic field only in finite space field line F = 0 along axis closes at infinity field lines on circular tori whose cross section are the lines F = const z F = 0 F = const z grad F = 0 or B = 0 p O-type neutral point s O-type neutral line axisymmetry: closed neutral line around neutral line is B 0 j ϕ 0, but there is no source of j ϕ : E ϕ = 0 because of axisymmetry and (u B) ϕ = 0 on neutral line for finite u

17 Cowling s theorem Formal proof Introduction Antidynamo theorems Parker s helical convection Mean-field theory Mean-field coefficients Mean-field dynamos Consider vicinity of neutral line, assume axisymmetry B p dl = B dl = B df = 4π j df = 4π j ϕ df c c = 4πσ u c 2 p B p df 4πσ u c 2 p B p df 4πσ c u 2 p,max integration circle of radius ε B p 2πε 4πσ c u p,maxb 2 p πε 2 or 1 2πσ c u p,maxε 2 ε 0 u p,max contradiction B p df

18 Toroidal theorems Introduction Antidynamo theorems Parker s helical convection Mean-field theory Mean-field coefficients Mean-field dynamos Toroidal velocity theorem (Elsasser 1947, Bullard & Gellman 1954) A toroidal motion in a spherical conductor can not maintain a magnetic field by dynamo action. Sketch of proof: d dt (r B) = η 2 (r B) for r u = 0 r B 0 for t P 0 T 0 Toroidal field theorem / Invisible dynamo theorem (Kaiser et al. 1994) A purely toroidal magnetic field can not be maintained by a dynamo.

19 Parker s helical convection Antidynamo theorems Parker s helical convection Mean-field theory Mean-field coefficients Mean-field dynamos velocity vorticity helicity u ω = u H = u ω (Parker 1955)

20 Mean-field theory Introduction Antidynamo theorems Parker s helical convection Mean-field theory Mean-field coefficients Mean-field dynamos Statistical consideration of turbulent helical convection on mean magnetic field (Steenbeck, Krause and Rädler 1966) B = (u B) η B t u = u + u, B = B + B Reynolds rules for averages B = (u B + E) η B t E = u B mean electromotive force B = (u B + u B + G) η B t G = u B u B usually neglected, FOSA = SOCA B linear, homogeneous functional of B approximation of scale separation: B depends on B only in small surrounding Taylor expansion: ( u B ) i = α ij B j + β ijk B k / x j +...

21 Mean-field theory Introduction Antidynamo theorems Parker s helical convection Mean-field theory Mean-field coefficients Mean-field dynamos ( u B ) = α ij B j + β ijk B k / x j +... i α ij and β ijk depend on u homogeneous, isotropic u : α ij = αδ ij, β ijk = βε ijk then u B = αb β B Ohm s law: j = σ(e + (u B)/c) j = σ(e + (u B)/c + (αb β B)/c) and c B = 4πj j = σ eff (E + (u B)/c + αb/c) B = (u B + αb) η eff B t with η eff = η + β Two effects: (1) α effect: j = σ eff αb/c (2) turbulent diffusivity: β η, η eff = β = η T

22 Antidynamo theorems Parker s helical convection Mean-field theory Mean-field coefficients Mean-field dynamos Sketch of dependence of α and β on u B = (u B + u B + G) η B t simplifying assumptions: G = 0, u incompressible, isotropic, u = 0, η = 0 B k = t t 0 ε klm ε mrs } {{ } δ kr δ ls δ ks δ lr E i = u B i = ε ijk u j (t) [ t x l (u r B s)dτ + B k (t 0) ( u k B l + u B l k t 0 x l x l t [ = ε ijk u (τ) j (t) u k B l u j t 0 x (t)u l (τ) } {{ l }} {{ } α β u ) ] l B k u B k l dτ + B k x l x (t 0) l B k x l ] dτ isotropic turbulence: α = 1 3 u u τ = 1 3 Hτ and β = 1 3 u 2 τ H helicity, τ correlation time

23 Introduction Antidynamo theorems Parker s helical convection Mean-field theory Mean-field coefficients Mean-field dynamos Mean-field coefficients derived from a MHD geodynamo simulation (Schrinner et al. 2007)

24 Mean-field dynamos Introduction Antidynamo theorems Parker s helical convection Mean-field theory Mean-field coefficients Mean-field dynamos Dynamo equation: B t spherical coordinates, axisymmetry u = (0, 0, Ω(r, ϑ)r sin ϑ) = (u B + αb η T B) B = (0, 0, B(r, ϑ, t)) + (0, 0, A(r, ϑ, t)) α, grad Ω B = r sin ϑ( A) Ω α 2 1 t A + η T 2 1 B A = αb + η T 2 1 t A with 2 1 = 2 (r sin ϑ) 2 B p B t rigid rotation has no effect no dynamo if α = 0 α term Ω term α 0 Ω L 2 { 1 α 2 dynamo with dynamo number R 2 α 1 αω dynamo with dynamo number R α R Ω α

25 Sketch of an αω dynamo Antidynamo theorems Parker s helical convection Mean-field theory Mean-field coefficients Mean-field dynamos Ω Ω Ω Ω out in αβ B poloidal field toroidal field by poloidal field toroidal field by differential rotation; by α-effect differential rotation; Ω/ r < 0 electric currents electric currents α cos θ by α-effect by α-effect periodically alternating field, here antisymmetric with respect to equator

26 Sketch of an α 2 dynamo Antidynamo theorems Parker s helical convection Mean-field theory Mean-field coefficients Mean-field dynamos B p B p Bt αb p αbt stationary field, here antisymmetric with respect to equator

27 Equations and parameters Proudman-Taylor theorem Convection in rotating sphere Taylor s constraint MHD equations of rotating fluids in non-dimensional form Navier-Stokes equation including Coriolis and Lorentz forces ( ) u E + u u 2 u + 2ẑ u + Π = Ra r T + 1 t r 0 Pm ( B) B Inertia Viscosity Coriolis Buoyancy Lorentz Induction equation B t = (u B) 1 Pm B Induction Diffusion Energy equation T t + u T = 1 Pr 2 T + Q Incompressibility and divergence-free magnetic field u = 0, B = 0

28 Non-dimensional parameters Equations and parameters Proudman-Taylor theorem Convection in rotating sphere Taylor s constraint Control parameters (Input) Parameter Definition Force balance Model value Earth value Rayleigh number Ra = αg 0 Td/νΩ buoyancy/diffusivity 1 50Ra crit Ra crit Ekman number E = ν/ωd 2 viscosity/coriolis Prandtl number Pr = ν/κ viscosity/thermal diff Magnetic Prandtl Pm = ν/η viscosity/magn. diff Diagnostic parameters (Output) Parameter Definition Force balance Model value Earth value Elsasser number Λ = B 2 /µρηω Lorentz/Coriolis Reynolds number Re = ud/ν inertia/viscosity < Magnetic Reynolds Rm = ud/η induction/magn. diff Rossby number Ro = u/ωd inertia/coriolis Earth core values: d m, u m s 1, ν 10 6 m 2 s 1

29 Proudman-Taylor theorem Equations and parameters Proudman-Taylor theorem Convection in rotating sphere Taylor s constraint Non-magnetic hydrodynamics in rapidly rotating system E 1, Ro 1 : viscosity and inertia small balance between Coriolis force and pressure gradient p = 2ρΩ u, : (Ω )u = 0 u z = 0 motion independent along axis of rotation, geostrophic motion (Proudman 1916, Taylor 1921) Ekman layer: At fixed boundary u = 0, violation of P.-T. theorem necessary for motion close to boundary allow viscous stresses ν 2 u for gradients of u in z-direction Ekman layer of thickness δ l E 1/2 L 0.2 m for Earth core

30 Equations and parameters Proudman-Taylor theorem Convection in rotating sphere Taylor s constraint Convection in rotating spherical shell inside tangent cylinder: g Ω: Coriolis force opposes convection outside tangent cylinder: P.-T. theorem leads to columnar convection cells exp(imϕ ωt) dependence at onset of convection, 2m columns which drift in ϕ-direction inclined outer boundary violates Proudman-Taylor theorem columns close to tangent cylinder around inner core inclined boundaries, Ekman pumping and inhomogeneous thermal buoyancy lead to secondary circulation along convection columns: poleward in columns with ω z < 0, equatorward in columns with ω z > 0 negative helicity north of the equator and positive one south

31 Equations and parameters Proudman-Taylor theorem Convection in rotating sphere Taylor s constraint Convection in rotating spherical shell H < 0 H > 0 ω z > 0 and < 0 cyclones / anticyclones Isosurfaces of positive (red) and negative (blue) vorticity ω z

32 Taylor s constraint Introduction 2ρΩ u = p + ρg + ( B) B/4π suggest that viscous and inertial effects may be small. If we set E = Ro = Equations and parameters obtain Proudman-Taylor theorem Convection in rotating sphere 1z V = qrat g + ( B) B. Taylor s constraint This is called the magnetostrophic approximation. It has a fundamental c discovered by Taylor (1963). If we take the φ-component of (3.36) Vs = + (( B) B)φ φ and integrate it over the surface of the cylinder C(s) (see figure 7) of radiu the rotation axis, we obtain Vs ds = (( B) B)φ ds. C(s) C(s) The cylinder C(s) intersects the outer sphere (r = 1) at z = zt,zb whe 1 s2 (=cos θ). The left-hand side of (3.38) is the net flow of fluid out o surface. For an incompressible fluid, and with no flow into or out of the end this must be zero, with the consequence that (( B) B)φ ds = 0. C(s) This is Taylor s constraint or Taylor s condition. magnetostrophic regime u = 0, ρ = const ; Ω = ω 0 e z Consider ϕ-component and integrate over cylindrical surface C(s) 2ρΩ C(s) p/ ϕ = 0 after integration over ϕ, g in meridional plane = 1 (( B) B) 4π ϕ ds u ds C(s) } {{ } = 0 C(s) (( B) B) ϕ ds = 0 (Taylor 1963) net torque by Lorentz force on any cylinder Ω vanishes Figure 7. The Taylor cylinder C(s), illustrated for the cases (a) where th the inner core, and (b) where s>rib. The cylinder extends from z = zt = rib 2 s2 and (b) zb = zt. From Fearn (1994). where (a) zb = B not necessarily small, but positive and negative parts of the integrand cancelling each other out The system has the freedom to satisfy this constraint through a co azimuthal flow that is otherwise undetermined. If we take the curl of V =0, we obtain violation by viscosity in Ekman boundary layers torsional V oscillations z around Taylor state = q Ra( T g) + (( B) B).

33 Benchmark dynamo Introduction A simple model Interpretation Advanced models Reversals Power requirement Ra = 10 5 = 1.8 Ra crit, E = 10 3, Pr = 1, Pm = 5 a) b) c) d) radial magnetic field radial velocity field axisymmetric axisymmetric at outer radius at r = 0.83r 0 magnetic field flow (Christensen et al. 2001)

34 A simple model Interpretation Advanced models Reversals Power requirement Conversion of toroidal field into poloidal field (Olson et al. 1999)

35 A simple model Interpretation Advanced models Reversals Power requirement Generation of toroidal field from poloidal field (Olson et al. 1999)

36 A simple model Interpretation Advanced models Reversals Power requirement Field line bundle in the benchmark dynamo c (cf Aubert)

37 Strongly driven dynamo model A simple model Interpretation Advanced models Reversals Power requirement Ra = = 42 Ra crit, E = , Pr = 1, Pm = 2.5 a) b) c) d) radial magnetic field radial velocity field axisymmetric axisymmetric at outer radius at r = 0.93r 0 magnetic field flow (Christensen et al. 2001)

38 Introduction Comparison of the radial magnetic field at the CMB a c GUFM model (Jackson et al. 2000) Full numerical simulation d b c A simple model Interpretation Advanced models Reversals Power requirement Spectrally filtered simulation at E = , Ra = 42 Racrit, Pm = 1, Pr = 1 Reversing dynamo at E = , Ra = 26 Racrit, Pm = 3, Pr = 1 (Christensen & Wicht 2007)

39 A simple model Interpretation Advanced models Reversals Power requirement Dynamical Magnetic Field Line Imaging / Movie 2 (Aubert et al. 2008) rom (a): DMFI movie 1 of model C and (b): movie 2 of model T. Left-hand panels: top view. Right-hand panels: sid ) boundaries of the model are colour-coded with Dieter the Schmitt radial Geomagnetic field Dynamo (a Theory red patch denotes outwards oriented field

40 Reversals Introduction A simple model Interpretation Advanced models Reversals Power requirement 500 years before midpoint midpoint 500 years after midpoint (Glatzmaier and Roberts 1995)

41 22:46 Geophysical Journal International Magnetohydrodynamical gji 3693 dynamos Reversals J. Aubert, J. Aurnou and J. Wicht A simple model Interpretation Advanced models Reversals Power requirement January 14, :46 Geophysical JournalInternational gji J. Aubert, J. Aurnou and J. Wicht petition between normal and inverse structures is felt at t well as at the surface, through an axial quadrupole mag As in event E1, the axial dipole is not efficiently mainta configuration, leaving mostly equatorial field lines insid maintained by two equatorial magnetic upwellings (Fig slightly inverse (red) e z orientation. Also similar to even petition between faint normal and inverse magnetic antic be observed (Fig. 12c) until time where inverse stru precedence and rebuild an axial dipole (Fig. 12d), thus the reversal sequence. The DMFI sequence for event E2 the role of magnetic upwellings in a scenario which is b sistent with that proposed by Sarson & Jones (1999). In the smaller-scale model C, the influence of m wellings on the dipole latitude and amplitude is not as in model T. Their appearance are, however, associated of the dipole axis as seen from the Earth s surface (see inserts in movie 1). Thus, we argue that two essential for the production of excursions and reversals in numeric are the existence of magnetic upwellings and a multipola netic field. This agrees with the models of Wicht & Ol in which the start of a reversal sequence was found to co upwelling events inside the tangent cylinder. (Aubert et al. 2008)

42 A simple model Interpretation Advanced models Reversals Power requirement Power requirement of the geodynamo Dynamo converts thermal and gravitational energy into magnetic energy Power requirement for geodynamo set by its ohmic losses Difficult to estimate: TW % Earth s surface heat flow Recent estimate from numerical and laboratory dynamos by extrapolating the magnetic dissipation time for realistic values of the control parameters: TW (Christensen and Tilgner 2004) Important for thermal budget and evolution of the inner core Thermal convection thermodynamically inefficient, compositional convection associated with core cooling High heat flow leads to rapid growth of the inner core and low age 1 Gyr Low power requirement estimated by Christensen and Tilgner (2004) is consistent with an inner core as old as 3.5 Gyr

43 Literature Introduction P. H. Roberts, An introduction to magnetohydrodynamics, Longmans, 1967 H. Greenspan, The theory of rotating fluids, Cambridge, 1968 H. K. Moffatt, Magnetic field generation in electrically conducting fluids, Cambridge, 1978 E. N. Parker, Cosmic magnetic fields, Clarendon, 1979 F. Krause, K.-H. Rädler, Mean-field electrodynamics and dynamo theory, Pergamon, 1980 F. Krause, K.-H. Rädler, G. Rüdiger, The cosmic dynamo, IAU Symp. 157, Kluwer, 1993 M. R. E. Proctor and A. D. Gilbert (Eds.), Lectures on solar and planetary dynamos, Cambridge, 1994 Treatise on geophysics, Vol. 8, Core dynamics, P. Olson (Ed.), Elsevier, 2007 P. H. Roberts, Theory of the geodynamo C. A. Jones, Thermal and compositional convection in the outer core U. R. Christensen and J. Wicht, Numerical dynamo simulations G. A. Glatzmaier and R. S. Coe, Magnetic polarity reversals in the core