Geomagnetic Dynamo Theory

Introduction Basic electrodynamics Kinematic turbulent dynamos MHD dynamos Geodynamo simulations

Dieter Schmitt

Max Planck Institute for Solar System Research

IMPRS Solar System School Retreat Travemünde-Brodten 26-30 April 2009

Dieter Schmitt Geomagnetic Dynamo Theory Introduction Basic electrodynamics Kinematic turbulent dynamos MHD dynamos Geodynamo simulations Outline

1 Introduction Mean-field theory Geomagnetic field Mean-field dynamos Dynamo hypothesis 4 MHD dynamos Homopolar dynamo Equations and parameters 2 Basic electrodynamics Proudman-Taylor theorem Pre-Maxwell theory Convection in rotating sphere Induction equation Taylor’s constraint Alfven’s theorem 5 Geodynamo simulations Magnetic Reynolds number A simple model Poloidal and toroidal fields Interpretation 3 Kinematic turbulent dynamos Advanced models Antidynamo theorems Reversals Parker’s helical convection 6 Literature

Dieter Schmitt Geomagnetic Dynamo Theory Introduction Basic electrodynamics Geomagnetic ﬁeld Kinematic turbulent dynamos Dynamo hypothesis MHD dynamos Homopolar dynamo Geodynamo simulations Geomagnetic ﬁeld

1600 Gilbert, De Magnete: ”Magnus magnes ipse est globus terrestris.” (The Earth’s globe itself is a great magnet.)

1838 Gauss: Mathematical description of geomagnetic ﬁeld l+1 P m − P ∇ m − P ∇ R m m m B = l,m Bl = Φl = R r Pl (cos ϑ) gl cos mφ + hl sin mφ sources inside Earth l number of nodal lines, m number of azimuthal nodal lines l = 1, 2, 3,... dipole, quadrupole, octupole, ... m = 0 axisymmetry, m = 1, 2,... non-axisymmetry 0 ≈ − | m| | m| Earth: g1 0.3 G, all other gl , hl < 0.05 G h i1/2 3 0 2 1 2 1 2 ≈ · 25 3 mainly dipolar, dipole moment µ = R (g1) + (g1) + (h1 ) 8 10 G cm h i1/2 . 1 2 1 2 0 ≈ ◦ tan ψ = (g1) + (h1 ) g1, dipole tilt ψ 11 dipole : quadrupole ≈ 1 : 0.14 (at CMB)

CMB

ICB

Br at surface 1990

= l

Br at CMB 1990

Br at CMB 1890

westward drift 0.18◦/yr u ≈ 0.3 mm/sec

Br at CMB 1990

SINT-800 VADM (Guyodo and Valet 1999)

NGP (Ohno and Hamano 1992)

Larmor (1919): Magnetic field of Earth and Sun maintained by self-excited dynamo Dynamo: u×B y j y B y u Faraday Ampere Lorentz motion of an electrical conductor in an ’inducing’ magnetic field y induction of electric current Self-excited dynamo: inducing magnetic field created by the electric current (Siemens 1867) Example: homopolar dynamo Homogeneous dynamo (no wires in Earth core or solar convection zone) y complex motion necessary Kinematic (u prescribed, linear) Dynamic (u determined by forces, including Lorentz force, non-linear)

u uxB

electromotive force u×B y electric current through wire loop y induced magnetic ﬁeld reinforces applied magnetic ﬁeld self-excitation if rotation Ω > 2πR/M is maintained where R resistance, M inductance

Dieter Schmitt Geomagnetic Dynamo Theory Introduction Pre-Maxwell theory Basic electrodynamics Induction equation Kinematic turbulent dynamos Alfven’s theorem MHD dynamos Magnetic Reynolds number Geodynamo simulations Poloidal and toroidal ﬁelds Pre-Maxwell theory

Maxwell equations: cgs system, vacuum, B = H, D = E ∂E ∂B c∇×B = 4πj + , c∇×E = − , ∇·B = 0 , ∇·E = 4πλ ∂t ∂t

Basic assumptions of MHD: • u c: system stationary on light travel time, no em waves • high electrical conductivity: E determined by ∂B/∂t, not by charges λ E B E 1 L u e E2 c ≈ y ≈ ≈ 1 , E plays minor role : el ≈ 1 2 L T B c T c em B ∂E/∂t E/T E u u2 ≈ ≈ ≈ 1 , displacement current negligible c∇×B cB/L B c c2

Pre-Maxwell equations: ∂B c∇×B = 4πj , c∇×E = − , ∇·B = 0 ∂t

Pre-Maxwell equations Galilei-covariant: 1 E0 = E + u×B , B0 = B , j0 = j c Relation between j and E by Galilei-covariant Ohm’s law: j0 = σE0 in resting frame of reference, σ electrical conductivity 1 j = σ(E + u×B) c

Magnetohydrokinematics: Magnetohydrodynamics: c∇×B = 4πj additionally ∂B c∇×E = − Equation of motion ∂t Equation of continuity ∇·B = 0 Equation of state 1 j = σ(E + u×B) Energy equation c Dieter Schmitt Geomagnetic Dynamo Theory Introduction Pre-Maxwell theory Basic electrodynamics Induction equation Kinematic turbulent dynamos Alfven’s theorem MHD dynamos Magnetic Reynolds number Geodynamo simulations Poloidal and toroidal ﬁelds Induction equation

Evolution of magnetic ﬁeld ! ! ∂B j 1 c 1 = −c∇×E = −c∇× − u×B = −c∇× ∇×B − u×B ∂t σ c 4πσ c ! c2 = ∇×(u×B) − ∇× ∇×B = ∇×(u×B) − η∇×∇×B 4πσ c2 with η = = const magnetic diffusivity 4πσ induction, diffusion

∇×(u×B) = −B ∇·u + (B·∇)u − (u·∇)B [ +u ∇·B = 0 ] expansion/contraction, shear/stretching, advection

∇·B = 0 as initial condition, conserved

∂B Ideal conductor η = 0 : = ∇×(u×B) ∂t d Z Magnetic ﬂux through ﬂoating surface is constant : B·dF = 0 dt F

(Alfvén 1942)

Frozen-in field lines: impression that magnetic field follows flow, but cE = −u×B and c∇×E = −∂B/∂t ∂B = ∇×(u×B) = −B ∇·u + (B·∇)u − (u·∇)B ∂t

(i) star contraction: B ∼ R−2, ρ ∼ R−3 y B ∼ ρ2/3 Sun y white dwarf y neutron star: ρ [g cm−3]: 1 y 106 y 1015 (ii) stretching of ﬂux tube: Bd2 = const, ld2 = const y B ∼ l (iii) shear, differential rotation

∂Bφ/∂t = r sin θ ∇Ω·Bp

Dimensionless variables: length L, velocity u0, time L/u0

∂B u L = ∇×(u×B) − R−1 ∇×∇×B with R = 0 ∂t m m η

as combined parameter

laboratorium: Rm 1, cosmos: Rm 1

induction for Rm 1, diffusion for Rm 1, e.g. for small L example: ﬂux expulsion from closed velocity ﬁelds

(Weiss 1966)

Spherical coordinates (r, ϑ, ϕ)

Axisymmetric ﬁelds: ∂/∂ϕ = 0

B(r, ϑ) = (B , B , B ) =0 r ϑ ϕ z}|{ 2 1 ∂r B 1 ∂ sin ϑB 1 ∂Bϕ ∇·B = 0 y r + ϑ + = 0 r2 ∂r r sin ϑ ∂ϑ r sin ϑ ∂ϕ

B = Bp + Bt poloidal and toroidal magnetic ﬁeld

Bt = (0, 0, Bϕ) satisﬁes ∇·Bt = 0

Bp = (Br , Bϑ, 0) = ∇×A with A = (0, 0, Aϕ) satisﬁes ∇·Bp = 0 ! 1 ∂r sin ϑAϕ ∂r sin ϑAϕ B = , − , 0 p r sin ϑ r∂ϑ ∂r axisymmetric magnetic ﬁeld determined by the two scalars: r sin ϑAϕ and Bϕ

Axisymmetric fields: c c j = ∇×B , j = ∇×B t 4π p p 4π t r sin ϑAϕ = const : field lines of poloidal field in meridional plane

ﬁeld lines of Bt are circles around symmetry axis

Non-axisymmetric ﬁelds:

B = Bp + Bt = ∇×∇×(Pr) + ∇×(Tr) = −∇×(r ×∇P) − r ×∇T r = (r, 0, 0) , P(r, ϑ, ϕ) and T(r, ϑ, ϕ) defining scalars c c ∇·B = 0 , j = ∇×B , j = ∇×B t 4π p p 4π t r ·Bt = 0 field lines of the toroidal field lie on spheres, no r component

Bp has in general all three components

Dieter Schmitt Geomagnetic Dynamo Theory Introduction Antidynamo theorems Basic electrodynamics Parker’s helical convection Kinematic turbulent dynamos Mean-ﬁeld theory MHD dynamos Mean-ﬁeld dynamos Geodynamo simulations Antidynamo theorems

Cowling’s theorem (Cowling 1934) Axisymmetric magnetic ﬁelds can not be maintained by a dynamo.

Toroidal velocity theorem (Elsasser 1947, Bullard & Gellman 1954) A toroidal motion in a spherical conductor can not maintain a magnetic ﬁeld by dynamo action.

Toroidal ﬁeld theorem / Invisible dynamo theorem (Kaiser et al. 1994) A purely toroidal magnetic ﬁeld can not be maintained by a dynamo.

velocity u

vorticity ω = ∇×u

helicity H = u·ω

(Parker 1955)

Statistical consideration of turbulent helical convection on mean magnetic ﬁeld (Steenbeck, Krause and Rädler 1966) ∂B = ∇×(u×B) − η∇×∇×B ∂t u = u + u0 , B = B + B0 Reynolds rules for averages ∂B = ∇×(u×B + E) − η∇×∇×B ∂t E = u0 ×B0 mean electromotive force ∂B0 = ∇×(u×B0 + u0 ×B + G) − η∇×∇×B0 ∂t G = u0 ×B0 − u0 ×B0 usually neglected , FOSA = SOCA B0 linear, homogeneous functional of B approximation of scale separation: B0 depends on B only in small surrounding 0 0 Taylor expansion: u ×B = αijBj + βijk ∂Bk /∂xj + ... i

0 0 u ×B = αijBj + βijk ∂Bk /∂xj + ... i 0 αij and βijk depend on u and are, in general, tensors 0 homogeneous, isotropic u : αij = αδij , βijk = −βεijk then u0 ×B0 = αB − β∇×B ∂B = ∇×(u×B + αB) − (η + β)∇×∇×B ∂t Two effects:

(1) α − effect: mean current parallel mean magnetic ﬁeld 1 1 α = − u0 ·∇×u0τ∗ = − Hτ∗ where H helicity , τ∗ correlation time 3 3 1 (2) turbulent diffusivity: β = u02τ∗ η , η + β = β = η 3 T

∂B Dynamo equation: = ∇×(u×B + αB − η ∇×B) ∂t T • spherical coordinates, axisymmetry • u = (0, 0, Ω(r, ϑ)r sin ϑ) • B = (0, 0, B(r, ϑ, t)) + ∇×(0, 0, A(r, ϑ, t)) α , grad Ω ∂B = r sin ϑ(∇×A)·∇Ω − α∇2A + η ∇2B ∂t 1 T 1 ∂A B p B = αB + η ∇2A with ∇2 = ∇2 − (r sin ϑ)−2 t ∂t T 1 1 rigid rotation has no effect α no dynamo if α = 0  2 2  1 α −dynamo with dynamo number Rα α−term α0  ≈  ∼ 1 α2Ω−dynamo ∇Ω−term |∇Ω|L 2   1 αΩ−dynamo with dynamo number RαRΩ

B out αΒ in 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

Bp Bp

αBp αBt

stationary ﬁeld, here antisymmetric with respect to equator

Dieter Schmitt Geomagnetic Dynamo Theory Introduction Equations and parameters Basic electrodynamics Proudman-Taylor theorem Kinematic turbulent dynamos Convection in rotating sphere MHD dynamos Taylor’s constraint Geodynamo simulations MHD equations of rotating ﬂuids in non-dimensional form

Navier-Stokes equation including Coriolis and Lorentz forces ! ∂u Ra E r 1 E + u·∇u − ∇2u + 2zˆ×u + ∇Π = T + (∇×B)×B ∂t Pr r0 Pm Inertia Viscosity Coriolis Buoyancy Lorentz

Induction equation ∂B 1 = ∇×(u×B) − ∇×∇×B ∂t Pm Induction Diffusion Energy equation ∂T 1 + u·∇T = ∇2T + Q ∂t Pr Incompressibility and divergence-free magnetic ﬁeld

∇·u = 0 , ∇·B = 0

Control parameters (Input) Parameter Deﬁnition Force balance Model value Earth value

Rayleigh number Ra = αg0∆Td/νκ buoyancy/diffusivity 1 − 50Racrit Racrit Ekman number E = ν/Ωd2 viscosity/Coriolis 10−6 − 10−4 10−14 Prandtl number Pr = ν/κ viscosity/thermal diff. 2·10−2 − 103 0.1 − 1 Magnetic Prandtl Pm = ν/η viscosity/magn. diff. 10−1 − 103 10−6 − 10−5

Diagnostic parameters (Output) Parameter Deﬁnition Force balance Model value Earth value Elsasser number Λ = B2/µρηΩ Lorentz/Coriolis 0.1 − 100 0.1 − 10 Reynolds number Re = ud/ν inertia/viscosity < 500 108 − 109 Magnetic Reynolds Rm = ud/η induction/magn. diff. 50 − 103 102 − 103 Rossby number Ro = u/Ωd inertia/Coriolis 3·10−4 − 10−2 10−7 − 10−6

Earth core values: d ≈ 2·105 m, u ≈ 2·10−4 m s−1, ν ≈ 10−6 m2s−1

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 = 0 motion independent along axis of rotation, geostrophic motion ∂z (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∇2u for gradients of u in z-direction 1/2 Ekman layer of thickness δl ∼ E L ∼ 0.2 m for Earth core

inside tangent cylinder: g k 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 y 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 ωz < 0, equatorward in columns with ωz > 0 y negative helicity north of the equator and positive one south

H < 0

H > 0

ωz > 0 and < 0 cyclones / anticyclones

Dieter Schmitt Geomagnetic Dynamo Theory Hydromagnetic ﬂow in planetary cores 199

3.3.3. Taylor’s constraint. The very low geophysical values of E and Ro (see section 2.2) Introduction suggest that viscous and inertial effects may be small. If we set E Ro 0 in (2.20), we Equations and parameters = = Basic electrodynamics obtain Proudman-Taylor theorem Kinematic turbulent dynamos Convection in rotating sphere 1z V 5 qRaT g ( B) B. (3.36) MHD dynamos × =−∇ − + ∇ × × Taylor’s constraint This is called the magnetostrophic approximation. It has a fundamental consequence first Geodynamo simulations discovered by Taylor (1963). If we take the φ-component of (3.36) ∂5 Vs (( B) B)φ (3.37) Taylor’s constraint =−∂φ + ∇ × × and integrate it over the surface of the cylinder C(s) (see figure 7) of radius s, coaxial with the rotation axis, we obtain × −∇ ∇× × 2ρΩ u = p + ρg + ( B) B/4π magnetostrophic regimeVs dS (( B) B)φ dS. (3.38) = ∇ × × ZC(s) ZC(s) ∇· The cylinder C(s) intersects the outer sphere (r 1) at z zT,zB where zT zB u = 0 , ρ = const ; Ω = ω0ez √1 s2( cos θ). The left-hand side of (3.38) is the= net flow= of fluid out of the cylindrical=− = surface.− For= an incompressible fluid, and with no flow into or out of the ends of the cylinder, Consider ϕ-component and integrate over cylindricalthis surface must be zero, withC the( consequences) that

(( B) B)φ dS 0. (3.39) ∇ × × = ∂p/∂ϕ = 0 after integration over ϕ, g in meridionalZ planeC(s) This is Taylor’s constraint or Taylor’s condition. Z 1 Z 2ρΩ u·ds = ((∇×B)×B)ϕ ds C(s) 4π C(s) | {z } = 0 Z ((∇×B)×B)ϕ ds = 0 (Taylor 1963) C(s)

net torque by Lorentz force on any cylinder k Ω vanishes Figure 7. The Taylor cylinder C(s), illustrated for the cases (a) where the cylinder intersects 2 the inner core, and (b) where s>rib. The cylinder extends from z zT √1 s to z zB, = = − = 2 2 where (a) zB r s and (b) zB zT. From Fearn (1994). = ib − =− B not necessarily small, but positive and negative parts of the integrandq The system has the freedom to satisfy this constraint through a component of the cancelling each other out azimuthal ﬂow that is otherwise undetermined. If we take the curl of (3.36) and use V 0, we obtain ∇· = ∂V violation by viscosity in Ekman boundary layers y torsional oscillationsq Ra( T g) (( B) B). (3.40) − ∂z = ∇ × + ∇ × ∇ × × around Taylor state Dieter Schmitt Geomagnetic Dynamo Theory Introduction A simple model Basic electrodynamics Interpretation Kinematic turbulent dynamos Advanced models MHD dynamos Reversals Geodynamo simulations Benchmark dynamo

5 −3 Ra = 10 = 1.8 Racrit , E = 10 , Pr = 1 , Pm = 5 a) b) c) d)

radial magnetic field radial velocity field axisymmetric axisymmetric at outer radius at r = 0.83r0 magnetic field flow

(Christensen et al. 2001)

Dieter Schmitt Geomagnetic Dynamo Theory Introduction A simple model Basic electrodynamics Interpretation Kinematic turbulent dynamos Advanced models MHD dynamos Reversals Geodynamo simulations Conversion of toroidal ﬁeld into poloidal ﬁeld a

(Olson et al. 1999) Dieter Schmitt Geomagnetic Dynamo Theory b

c a

Introduction A simple model Basic electrodynamics Interpretation Kinematic turbulent dynamos Advanced models MHD dynamos Reversals Geodynamo simulations Generation of toroidal ﬁeld from poloidal ﬁeld b

(Olson et al. 1999) Dieter Schmitt Geomagnetic Dynamo Theory c a

Introduction A simple model Basic electrodynamics Interpretation Kinematic turbulent dynamos Advanced models MHD dynamos Reversals Geodynamo simulations Field line bundle in the benchmark dynamo c

(cf Aubert)

8 −5 Ra = 1.2×10 = 42 Racrit , E = 3×10 , Pr = 1 , Pm = 2.5 a) b) c) d)

radial magnetic field radial velocity field axisymmetric axisymmetric at outer radius at r = 0.93r0 magnetic field flow

(Christensen et al. 2001)

Dieter Schmitt Geomagnetic Dynamo Theory a

Introduction A simple model Basic electrodynamics Interpretation Kinematic turbulent dynamos Advanced models MHD dynamos Reversals Geodynamo simulations Comparison of the radial magnetic ﬁeld at the CMB a c

GUFM model Full numerical (Jackson et al. 2000) simulation b d

Spectrally ﬁltered simulation at Reversing dynamo at −5 −4 c E = 3·10 , Ra = 42 Racrit, Pm = 1, Pr = 1 E = 3·10 , Ra = 26 Racrit, Pm = 3, Pr = 1 (Christensen & Wicht 2007) Dieter Schmitt Geomagnetic Dynamo Theory

d January 14, 2008 22:46 GeophysicalJournalInternational gji˙3693

Magnetic structure of numerical dynamos 5

Introduction A simple model Basic electrodynamics Interpretation Kinematic turbulent dynamos Advanced models MHD dynamos Reversals Geodynamo simulations Dynamical Magnetic Field Line Imaging / Movie 2

(Aubert et al. 2008) Figure 5. Snapshots from (a): DMFI movie 1 of model C and (b): movie 2 of model T. Left-hand panels: top view. Right-hand panels: side view. The inner (ICB) and outer (CMB) boundaries of the model are colour-coded withDieter the Schmitt radial magneticGeomagnetic field Dynamo (aTheory red patch denotes outwards oriented field). In addition, the outer boundary is made selectively transparent, with a transparency level that is inversely proportional to the local radial magnetic field. Field lines are also colour-coded in order to indicate ez-parallel (red) and antiparallel (blue) direction. The radial magnetic field as seen from the Earth’s surface is represented in the upper-right inserts, in order to keep track of the current orientation and strength of the large-scale magnetic dipole. Colour maps for (a): ICB field from −0.12 (blue) to 0.12 (red), in units of (ρμ)1/2D, CMB field from −0.03 to 0.03, Earth’s surface field from −210−4 to210−4. For (b): ICB field from −0.72 to 0.72, CMB field from −0.36 to 0.36, Earth’s surface field from −1.8 10−3 to 1.8 10−3.

vortices into columns elongated along the ez axis of rotation, due to the Proudman–Taylor constraint. The sparse character of the mag- 3.1.1 Magnetic cyclones netic energy distribution results from the tendency of field lines A strong axial flow cyclone (red isosurface in Fig. 4) winds and to cluster at the edges of flow vortices due to magnetic field ex- stretches field lines to form a magnetic cyclone. Fig. 6 relates DMFI pulsion (Weiss 1966; Galloway & Weiss 1981). Since magnetic visualizations of magnetic cyclones, as displayed in Figs 4 and 5, field lines correlate well with the flow structures in our models, with a schematic view inspired by Olson et al. (1999). A mag- we will subsequently visualize the magnetic field structure alone. netic cyclone can be identified by the anticlockwise motion of field The supporting movies of this paper (see Fig. 1 for time window lines clustered close to the equator, moving jointly with fairly stable and Figs 5–9 for extracts) present DMFI field lines, together with high-latitude CMB flux patches concentrated above and below the radial magnetic flux patches at the inner boundary (which we will centre of the field line cluster. Model C (movie 1, Fig. 5a) exhibits refer to as ICB) and the selectively transparent outer boundary very large-scale magnetic cyclones (times 4.3617, 4.3811), which (CMB). We will first introduce the concept of a magnetic vortex, suggest an axial vorticity distribution biased towards flow cyclones. which is defined as a field line structure resulting from the inter- Inside these vortices, the uneven distribution of buoyancy along ez action with a flow vortex. By providing illustrations of magnetic creates a thermal wind secondary circulation (Olson et al. 1999), cyclones and anticyclones, DMFI provides a dynamic, field-line which is represented in red on Fig. 6. This secondary circulation based visual confirmation of previously published dynamo mech- concentrates CMB flux at high latitudes, giving rise to relatively anisms (Kageyama & Sato 1997; Olson et al. 1999; Sakuraba & long-lived (several vortex turnovers) flux patches similar to those Kono 1999; Ishihara & Kida 2002), and allows the extension of found in geomagnetic field models. Simultaneously, close to the such descriptions to time-dependent, spatially complex dynamo equatorial plane, the secondary circulation concentrates field lines regimes. into bundles and also pushes them towards the outer boundary,where

C 2008 The Authors, GJI Journal compilation C 2008 RAS Introduction A simple model Basic electrodynamics Interpretation Kinematic turbulent dynamos Advanced models MHD dynamos Reversals Geodynamo simulations Reversals

500 years before midpoint midpoint 500 years after midpoint

(Glatzmaier and Roberts 1995)

Dieter Schmitt Geomagnetic Dynamo Theory January 14, 2008 22:46 GeophysicalJournalInternational gji˙3693

10 J. Aubert, J. Aurnou and J. Wicht

petition between normal and inverse structures is felt at the CMB, as well as at the surface, through an axial quadrupole magnetic field. As in event E1, the axial dipole is not efficiently maintained by this configuration, leaving mostly equatorial field lines inside the shell, maintained by two equatorial magnetic upwellings (Fig. 12b), with slightly inverse (red) ez orientation. Also similar to event E1, a com- petition between faint normal and inverse magnetic anticyclones can be observed (Fig. 12c) until time 171.5 where inverse structures take precedence and rebuild an axial dipole (Fig. 12d), thus completing the reversal sequence. The DMFI sequence for event E2 highlights the role of magnetic upwellings in a scenario which is broadly con- sistent with that proposed by Sarson & Jones (1999). In the smaller-scale model C, the influence of magnetic upwellings on the dipole latitude and amplitude is not as clear-cut as in model T. Their appearance are, however, associated with tilting Introduction A simple model Basic electrodynamics of the dipole axis as seen from the Earth’s surface (see upper-right Interpretation Kinematic turbulent dynamos Advanced models inserts in movie 1). Thus, we argue that two essential ingredients January 14, 2008 22:46 GeophysicalJournalInternational gMHDji˙3693 dynamos Reversals Geodynamo simulations for the production of excursions and reversals in numerical dynamos are the existence of magnetic upwellings and a multipolar ICB mag- 10 J.Reversals Aubert, J. Aurnou and continued J. Wicht netic field. This agrees with the models of Wicht & Olson (2004), in which the start of a reversal sequence was found to correlate with petition between normal and inverse structures is felt at the CMB,upwelling as events inside the tangent cylinder. well as at the surface, through an axial quadrupole magnetic field. As in event E1, the axial dipole is not efficiently maintained by this configuration, leaving mostly equatorial field lines inside the shell, maintained by two equatorial magnetic upwellings (Fig. 12b),4 with DISCUSSION slightly inverse (red) ez orientation. Also similar to event E1, a com- petition between faint normal and inverse magnetic anticyclonesUnderstanding can the highly complex processes of magnetic field gen- be observed (Fig. 12c) until time 171.5 where inverse structureseration take in the Earth’s core is greatly facilitated by Alfvén’s the- precedence and rebuild an axial dipole (Fig. 12d), thus completingorem and the frozen-flux approximation, provided one supplies the reversal sequence. The DMFI sequence for event E2 highlightsan imaging method which is adapted to the intrinsically 3-D and the role of magnetic upwellings in a scenario which is broadlytime-dependent con- nature of the problem, and also takes into account sistent with that proposed by Sarson & Jones (1999). diffusive effects. The DMFI technique used in the present study aims In the smaller-scale model C, the influence of magneticat up-achieving this goal, and highlights several magnetic structures: wellings on the dipole latitude and amplitude is not as clear-cutmagnetic as anticyclones are found outside axial flow anticyclones, in model T. Their appearance are, however, associated with tiltingand regenerate the axial dipole through the creation of magnetic of the dipole axis as seen from the Earth’s surface (see upper-rightloops characteristic of an alpha-squared dynamo mechanism. Mag- inserts in movie 1). Thus, we argue that two essential ingredientsnetic cyclones are found outside axial flow cyclones, and concentrate for the production of excursions and reversals in numerical dynamosthe magnetic flux into bundles where significant Ohmic dissipation are the existence of magnetic upwellings and a multipolar ICBtakes mag- place. Our description of magnetic vortices confirms and illus- netic field. This agrees with the models of Wicht & Olson (2004),trates previously published mechanisms, as presented for instance by in which the start of a reversal sequence was found to correlateOlson with et al. (1999). By separating the influence of cyclones and anticyclones, we extend these views to more complex cases where there Figure 12. Theupwelling steps involved events in insideevent E2, the(Aubert a tangentfull reversal cylinder. et of al.the dipole 2008) axis occurring at time 171.32 in movie 2 (model T). is a broken symmetry between cyclonic and anticyclonic motion. Dieter Schmitt Geomagnetic Dynamo Theory Furthermore, we present the first field line dynamic descriptions of magnetic upwellings, which are created by field line stretching and equatorial field lines of inverse (red) polarity. In this context, faint advection inside flow upwellings. magnetic anticyclones4 DISCUSSION producing poloidal field lines of both po- Our models show that the magnetic structures are robust features larities can beUnderstanding observed, which the highly do not complex have aprocesses net effect of onmagnetic the fieldfound gen- at high (model T) as well as moderately low (model C) values regenerationeration of the axial in the dipole, Earth’s which core in is turn greatly collapses. facilitated The equa- by Alfvén’sof the- the Ekman number. This suggests that they pertain to the Earth’s torial dipoleorem component and the is frozen-flux also bound approximation, to collapse due provided to the in- one suppliescore. Since we only have access to the radial component of the termittent characteran imaging of the method upwellings which iswhich adapted maintain to the it. intrinsically A low 3-Dmagnetic and field at the Earth’s CMB, our description of the magnetic amplitude multipolartime-dependent state, therefore, nature of takes the problem, place in the and whole also takes shell, into accountstructure underlying CMB flux patches in the models is of particular where againdiffusive faint magnetic effects. anticyclones The DMFI technique of both polarities used in the can present be studyinterest. aims Inside the tangent cylinder, short-lived CMB patches of seen at differentat achieving locations this (Fig. goal, 11c). and After highlights time 168.71 several the magnetic normal structures:both polarities can be created by the expulsion of azimuthal flux polarity (blue)magnetic magnetic anticyclones anticyclones are take found precedence, outside axial and regen- flow anticyclones,within a magnetic upwelling. These patches are quickly weakened erate an axialand dipole regenerate in 0.2 the magnetic axial dipole diffusion through times the (Fig. creation 11d). of magneticby the diverging flow on the top of the upwelling, therefore, they do Event E2loops starts characteristicwith two equatorial of an alpha-squared magnetic upwellings, dynamo grow- mechanism.not Mag- last more than a vortex turnover, which is equivalent to 60–300 yr ing from ICBnetic flux cyclones spots of opposite are found polarity, outside ataxial the flow edges cyclones, of adjacent and concentratein the Earth’score (Aubert et al. 2007). The observation of a tangent axial vorticesthe (Fig. magnetic 12a). The flux blue into upwelling bundles where feeds a significant normal polarity Ohmic dissipationcylinder inverse flux patch in the present geomagnetic field (Olson (blue) magnetictakes anticyclone, place. Our description while the red of magnetic upwelling vortices feeds anconfirms in- and& illus- Aurnou 1999; Jackson et al. 2000; Hulot et al. 2002), although it verse polaritytrates (red) previously magnetic anticyclone. published mechanisms, At time 171.327 as presented this com- for instanceis weakly by constrained and not observed with all field regularizations Olson et al. (1999). By separating the influence of cyclones and anticyclones, we extend these views to more complex cases where there C 2008 The Authors, GJI Figure 12. The steps involved in event E2, a full reversal of the dipole axis Journal compilation C 2008 RAS occurring at time 171.32 in movie 2 (model T). is a broken symmetry between cyclonic and anticyclonic motion. Furthermore, we present the first field line dynamic descriptions of magnetic upwellings, which are created by field line stretching and equatorial field lines of inverse (red) polarity. In this context, faint advection inside flow upwellings. magnetic anticyclones producing poloidal field lines of both po- Our models show that the magnetic structures are robust features larities can be observed, which do not have a net effect on the found at high (model T) as well as moderately low (model C) values regeneration of the axial dipole, which in turn collapses. The equa- of the Ekman number. This suggests that they pertain to the Earth’s torial dipole component is also bound to collapse due to the in- core. Since we only have access to the radial component of the termittent character of the upwellings which maintain it. A low magnetic field at the Earth’s CMB, our description of the magnetic amplitude multipolar state, therefore, takes place in the whole shell, structure underlying CMB flux patches in the models is of particular where again faint magnetic anticyclones of both polarities can be interest. Inside the tangent cylinder, short-lived CMB patches of seen at different locations (Fig. 11c). After time 168.71 the normal both polarities can be created by the expulsion of azimuthal flux polarity (blue) magnetic anticyclones take precedence, and regen- within a magnetic upwelling. These patches are quickly weakened erate an axial dipole in 0.2 magnetic diffusion times (Fig. 11d). by the diverging flow on the top of the upwelling, therefore, they do Event E2 starts with two equatorial magnetic upwellings, grow- not last more than a vortex turnover, which is equivalent to 60–300 yr ing from ICB flux spots of opposite polarity, at the edges of adjacent in the Earth’score (Aubert et al. 2007). The observation of a tangent axial vortices (Fig. 12a). The blue upwelling feeds a normal polarity cylinder inverse flux patch in the present geomagnetic field (Olson (blue) magnetic anticyclone, while the red upwelling feeds an in- & Aurnou 1999; Jackson et al. 2000; Hulot et al. 2002), although it verse polarity (red) magnetic anticyclone. At time 171.327 this com- is weakly constrained and not observed with all field regularizations

Dipole moment a

Time (Hoyng, Ossendrijver & Schmitt 2001)

Dieter Schmitt Geomagnetic Dynamo Theory Introduction Basic electrodynamics Kinematic turbulent dynamos MHD dynamos Geodynamo simulations Literature

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 (Eds.), 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 D. R. Fearn, Hydromagnetic flows in planetary cores, Rep. Prog. Phys., 61, 175, 1998

Dieter Schmitt Geomagnetic Dynamo Theory Introduction Basic electrodynamics Kinematic turbulent dynamos MHD dynamos Geodynamo simulations Literature continued

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 M. Ossendrijver, The solar dynamo, Astron. Astrophys. Rev., 11, 287, 2003 P. Charbonneau, Dynamo models of the solar cycle, Living Rev. Solar Phys., 2, 2005

Dieter Schmitt Geomagnetic Dynamo Theory