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Geomagnetic Dynamo Theory Introduction Basic electrodynamics Kinematic turbulent dynamos MHD dynamos Geodynamo simulations Geomagnetic Dynamo Theory 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) Dieter Schmitt Geomagnetic Dynamo Theory Introduction Basic electrodynamics Geomagnetic field Kinematic turbulent dynamos Dynamo hypothesis MHD dynamos Homopolar dynamo Geodynamo simulations Internal structure of the Earth CMB ICB Dieter Schmitt Geomagnetic Dynamo Theory Introduction Basic electrodynamics Geomagnetic field Kinematic turbulent dynamos Dynamo hypothesis MHD dynamos Homopolar dynamo Geodynamo simulations Spatial structure of geomagnetic field Br at surface 1990 = l Br at CMB 1990 Dieter Schmitt Geomagnetic Dynamo Theory Introduction Basic electrodynamics Geomagnetic field Kinematic turbulent dynamos Dynamo hypothesis MHD dynamos Homopolar dynamo Geodynamo simulations Secular variation Br at CMB 1890 westward drift 0.18◦/yr u ≈ 0.3 mm/sec Br at CMB 1990 Dieter Schmitt Geomagnetic Dynamo Theory Introduction Basic electrodynamics Geomagnetic field Kinematic turbulent dynamos Dynamo hypothesis MHD dynamos Homopolar dynamo Geodynamo simulations Secular variation continued SINT-800 VADM (Guyodo and Valet 1999) NGP (Ohno and Hamano 1992) Dieter Schmitt Geomagnetic Dynamo Theory Introduction Basic electrodynamics Geomagnetic field Kinematic turbulent dynamos Dynamo hypothesis MHD dynamos Homopolar dynamo Geodynamo simulations Polarity reversals Dieter Schmitt Geomagnetic Dynamo Theory Introduction Basic electrodynamics Geomagnetic field Kinematic turbulent dynamos Dynamo hypothesis MHD dynamos Homopolar dynamo Geodynamo simulations Dynamo hypothesis 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) Dieter Schmitt Geomagnetic Dynamo Theory Introduction Basic electrodynamics Geomagnetic field Kinematic turbulent dynamos Dynamo hypothesis MHD dynamos Homopolar dynamo Geodynamo simulations Homopolar dynamo B 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 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 continued 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 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 Alfven’s theorem ∂B Ideal conductor η = 0 : = ∇×(u×B) ∂t d Z Magnetic flux through floating surface is constant : B·dF = 0 dt F (Alfvén 1942) 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 Alfven’s theorem continued 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 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 Differential rotation ∂Bφ/∂t = r sin θ ∇Ω·Bp 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 Magnetic Reynolds number 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 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 Flux expulsion (Weiss 1966) 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 Poloidal and toroidal magnetic fields 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 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 Poloidal and toroidal magnetic fields continued 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
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