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Lecture 2: Dynamo Theory

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Lecture 2: Dynamo Theory Lecture 2: Dynamo theory Celine´ Guervilly School of Mathematics, Statistics and Physics, Newcastle University, UK Basis of electromagnetism: Maxwell’s equations Faraday’s law of induction: if a magnetic field B varies with time then an electric field E is produced. @B r × E = - @t Ampere’s` law (velocity speed of light) r × B = µ0j where j is the current density and µ0 is the vacuum magnetic permeability. Gauss’s law (electric monopoles from which electric field originates) ρ r · E = 0 with ρ the charge density and 0 the dielectric constant. No magnetic monopole (no particle from which magnetic field lines radiate) r · B = 0 Ohm’s law Relates current density j to electric field E. In a material at rest, we assume the simple form j = σE with σ the electrical conductivity. In the reference frame moving with the fluid, the electric, magnetic fields and current become E0 = E + u × B, B0 = B, j0 = j In the original reference, Ohm’s law is j = σ(E + u × B) A conducting wire is wound around the disc and joins the rim and the axis: electric current flows in the wire and across the disc. Winding is such that the induced magnetic field B reinforces the applied magnetic field B0. Dynamo: conversion of kinetic energy into magnetic energy. If the disc rotation rate exceeds a critical value, B0 can be switched off and the dynamo will continue to operate: the dynamo has become self-excited. Homopolar disc dynamo A solid electrically conducting disk rotates about an axis. Uniform magnetic field B0 aligned with the rotation axis. The electromotive force q(u × B0) separates the charges. Dynamo: conversion of kinetic energy into magnetic energy. If the disc rotation rate exceeds a critical value, B0 can be switched off and the dynamo will continue to operate: the dynamo has become self-excited. Homopolar disc dynamo A solid electrically conducting disk rotates about an axis. Uniform magnetic field B0 aligned with the rotation axis. The electromotive force q(u × B0) separates the charges. A conducting wire is wound around the disc and joins the rim and the axis: electric current flows in the wire and across the disc. Winding is such that the induced magnetic field B reinforces the applied magnetic field B0. If the disc rotation rate exceeds a critical value, B0 can be switched off and the dynamo will continue to operate: the dynamo has become self-excited. Homopolar disc dynamo A solid electrically conducting disk rotates about an axis. Uniform magnetic field B0 aligned with the rotation axis. The electromotive force q(u × B0) separates the charges. A conducting wire is wound around the disc and joins the rim and the axis: electric current flows in the wire and across the disc. Winding is such that the induced magnetic field B reinforces the applied magnetic field B0. Dynamo: conversion of kinetic energy into magnetic energy. Homopolar disc dynamo A solid electrically conducting disk rotates about an axis. Uniform magnetic field B0 aligned with the rotation axis. The electromotive force q(u × B0) separates the charges. A conducting wire is wound around the disc and joins the rim and the axis: electric current flows in the wire and across the disc. Winding is such that the induced magnetic field B reinforces the applied magnetic field B0. Dynamo: conversion of kinetic energy into magnetic energy. If the disc rotation rate exceeds a critical value, B0 can be switched off and the dynamo will continue to operate: the dynamo has become self-excited. Now using Faraday’s and Ampere’s` laws, we get @B = r × (u × B - ηr × B), @t where u is the fluid velocity, and η = 1=(µ0σ) is the magnetic diffusivity. If η is constant, then @B = r × (u × B) + ηr2B @t where we used r · B = 0. Magnetic induction equation Taking the curl of Ohm’s law, j r × = r × (E + u × B) σ If η is constant, then @B = r × (u × B) + ηr2B @t where we used r · B = 0. Magnetic induction equation Taking the curl of Ohm’s law, j r × = r × (E + u × B) σ Now using Faraday’s and Ampere’s` laws, we get @B = r × (u × B - ηr × B), @t where u is the fluid velocity, and η = 1=(µ0σ) is the magnetic diffusivity. Magnetic induction equation Taking the curl of Ohm’s law, j r × = r × (E + u × B) σ Now using Faraday’s and Ampere’s` laws, we get @B = r × (u × B - ηr × B), @t where u is the fluid velocity, and η = 1=(µ0σ) is the magnetic diffusivity. If η is constant, then @B = r × (u × B) + ηr2B @t where we used r · B = 0. Magnetic induction equation and magnetic Reynolds number For an incompressible fluid (r · u = 0) we can re-write the magnetic induction equation as DB = (B · r)u + ηr2B Dt where the material derivative is D @ ≡ + u · r Dt @t Induction is produced by the shear of magnetic field lines. The induction equation is linear in B and the relative importance of each term is independent of the field strength. Magnetic Reynolds number: jr × (u × B)j UL L2/η magnetic diffusion timescale ∼ = = jηr2Bj η L=U turnover timescale UL Rm ≡ η Magnetic diffusion is dominant on small scales (Rm 1) and negligible on large scales (Rm 1). Frozen flux limit: Rm ! Neglecting diffusion on the large scales: 1 DB = (B · r)u. Dt Magnetic field lines behave as material lines and move with the fluid. The magnetic flux through a material surface is conserved. B · dS = const. S This is Alfv´en’s theorem: Magnetic" flux is “frozen” into the fluid. If the surface expands then the field must become weaker, so that the total flux is unchanged. Alfven´ theorem no longer holds exactly if there is diffusion: reconnection is possible and field lines are no longer material curves. Poloidal-Toroidal decomposition r · B = 0 implies that only two independent scalar fields are needed to specify B. In spherical geometry: B = BT + BP, BT = r × Tr, BP = r × r × Pr. T: toroidal component, P: poloidal component. 1 B = L2(P), r r 1 @T 1 @ @rP Bθ = + , sin θ ∂φ r ∂θ @r @T 1 @ @rP Bφ = - + , ∂θ r sin θ ∂φ @r with @ @ 1 @ @ 1 @2 L2 = r2 - r2r2 = - sin θ - . @r @r sin θ ∂θ ∂θ sin2 θ @2φ Radial component of the induction equation gives the equation for P (L2P = r · B). Curl of the induction equation gives the equation for T (L2T = r · r × B). Radial component of the diffusion term and its curl can be separated into poloidal and toroidal parts. Only coupling arises from the induction term. Free decay modes: Rm ! 0 Consider a conducting sphere of radius a surrounded by an electrically insulating region. For r < a, @B = ηr2B, @t -σt We look for solutions of the form B = B0(r, θ, φ)e where σ is the decay rate. 2 We must solved the problem: (r + σ/η)B0 = 0. Spherical harmonics expansion: l l m m m m T = tl (r)Yl (θ, φ), P = pl (r)Yl (θ, φ). 1 1 = = Xl=0 mX0 Xl=0 mX0 2 2 m m This greatly simplifies the L operator: L (Yl ) = l(l + 1)Yl . For each spherical harmonics, we have to solve: 1 @ @pl σ l(l + 1) r2 t + - pl = 0 r2 @r @r η r2 t The solution of this equation is a spherical Bessel function jl. Free decay toroidal modes Outside the sphere (r > a): j = 0. For the toroidal component at r > a: 1 1 2 jr = r · r × B = L T = 0 ! T = 0. µ0 µ0 m p The toroidal decay solution for r < a is tl = jl( σ/ηr). 2 2 T is continuous at r = a so matching the solutions at r = a gives σl = ηxl =a , with xl the lowest zero of jl (for l = 1, x1 = 4.493). The decay time for the spherical harmonic of degree 2 2 l is τl = a /ηxl . In the Earth’s core (a =3500 km, η = 1m2/s), the toroidal mode l = 1 decays in about 20000 yrs. Toroidal modes of larger l decay faster. Free decay poloidal modes 2 For the poloidal component at r > a: jT = -r P/µ0 = 0 (jT: toroidal current). At r > a: A pm = , A = const. l rl+1 and so, @pm l + 1 l + pm = 0. @r r l m p The poloidal decay solution for r < a is pl = jl( σ/ηr). P and @P=@r are continuous at r = a so matching the solutions at r = a and using 2 2 the Bessel functions recurrence relations gives σl = ηxl-1=a . For the dipole l = 1, the lowest zero is x0 = π so the decay time for the dipole is τ = a2/ηπ2. In the Earth’s core, the dipole decays in about 40000 yrs. Poloidal modes with larger l decay faster. Magnetic Reynolds number in the core The turnover timescale is R 106m τ = ≈ = 60 yr U U 5 × 10-4mm/s τη 40000 yr Rm = ≈ ≈ 600 τU 60 yr Frozen flux approximation in Earth’s core? Core flow inversion Cowling’s anti-dynamo theorem Cowling (1934) Axisymmetric magnetic fields cannot be maintained by dynamo action. Important note: This theorem disallows axisymmetric B, but not axisymmetric u. An axisymmetric u can create a non-axisymmetric B. See examples of the Ponomarenko dynamo and Dudley & James dynamo. What about Saturn’s magnetic field then? Saturn’s magnetic field measured outside of the planet is remarkably axisymmetric..
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