Scope of this lecture ASTR 7500: Solar & Stellar Magnetism. Lecture 9 Tues 19 Feb Magnetic fields in the Universe. Geomagnetism.

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1 Scope of this lecture ASTR 7500: Solar & Stellar Magnetism Hale CGEG Solar & Space Physics Processes of magnetic field generation and destruction in turbulent plasma flows Introduction to general concepts of dynamo theory (this is not a lecture about the solar dynamo!) Outline Intro: Magnetic fields in the Universe MHD, induction equation Some general remarks and definitions regarding dynamos Small scale dynamos Large scale dynamos (mean field theory) Kinematic theory Characterization of possible dynamos Non-kinematic effects Matthias Rempel, Prof. Juri Toomre + HAO/NSO colleagues Lecture 9 Tues 19 Feb 2013 Concluding remarks zeus.colorado.edu/astr7500-toomre 1 / 74 Magnetic fields in the Universe 2 / 74 Geomagnetism Mostly dipolar field structure (currently) Earth Field strength 0.5G Magnetic field present for years, much longer than Ohmic decay time ( 104 years) Strong variability on shorter time scales (103 years) Mercury, Ganymede, (Io), Jupiter, Saturn, Uranus, Neptune have large scale fields Sun Magnetic fields from smallest observable scales to size of sun 22 year cycle of large scale field Ohmic decay time 109 years (in absence of turbulence) Other stars Stars with outer convection zone: similar to sun Stars with outer radiation zone: primordial fields, field generation in convective core Galaxies Field strength µg Field structure coupled to observed matter distribution Credit: NOAA NGDC 4 / 74 3 / 74 Geomagnetism Geomagnetism Long-term variation on scales of thousands to millions of years (deduced from volcanic rocks and sediments) Short-term variation on scales of hundreds of years Independent movement of the poles Mostly random changes of polarity South and North pole are in general not opposite to each other (higher multipoles) A given polarity for 100, 000 years Movements up to 40 km/year ( 1 mm/sec) Strong variation of dipole moment and failed reversals Fast switches 1000 years Credit: Arnaud Chulliat (Institut de Physique du Globe de Paris) Credit: US Geological Survey 5 / 74 6 / 74

2 Solar Magnetism Solar Magnetism Up to 4kG (sunspot umbra) field in solar photosphere Structured over the full range of observable scales from 100 km to size of Sun Large scale field shows symmetries with respect to equator Movie Full disk magnetogram SDO/HMI Large scale field exhibits 22 year magnetic cycle 11 year cycle present in large scale flow variations (meridional flow and differential rotation) 8 / 74 7 / 74 Solar Magnetism Galactic magnetism Magnetic field derived from polarization of radio emission µg field strength Magnetic field follows spiral structure to some extent Credit: NASA Optically thin dynamo Dynamo region can be observed! Cycle interrupted by grand minima with duration of up to 100 years Similar overall activity has been present for past 100, 000 years (tree ring and ice core records of cosmogenic isotopes: C-14 and Be-10). M51, Credit: MPI for Radioastronomy, Germany 9 / 74 Magnetic fields in the Universe 10 / 74 MHD equations The full set of MHD equations combines the induction equation with the Navier-Stokes equations including the Lorentz-force: % v % e % Objects from size of a planet to galaxy clusters have large scale ( size of object) magnetic fields Physical properties of object differ substantially 1,000 km to 100,000 LJ liquid iron to partially ionized plasma spherical to disk-shaped varying influence of rotation (but all of them are rotating) Rm = (%v) = %(v )v p + %g + 1 ( B) B + τ µ0 = %(v )e p v + (κ T ) + Qν + Qη = (v B η B) Assumptions: Validity of continuum approximation (enough particles to define averages) Non-relativistic motions, low frequencies Strong collisional coupling: validity of single fluid approximations, isotropic (scalar) gas pressure Is there a common origin of magnetic field in these objects? Can we understand this on basis of MHD? 11 / / 74

3 MHD equations Viscous stress tensor τ Λ ik = 1 ( vi + v ) k 2 x k x i ( τ ik = 2ϱν Λ ik 1 ) 3 δ ik v Ohmic dissipation Q η Equation of state Q ν = τ ik Λ ik, Q η = η µ 0 ( B) 2. p = ϱ e γ 1. ν, η and κ: viscosity, magnetic diffusivity and thermal conductivity µ 0 denotes the permeability of vacuum 13 / 74 Kinematic approach Solving the 3D MHD equations is not always feasible Semi-analytical approach preferred for understanding fundamental properties of dynamos Evaluate turbulent induction effects based on induction equation for a given velocity field Velocity field assumed to be given as background turbulence, Lorentz-force feedback neglected (sufficiently weak magnetic field) What correlations of a turbulent velocity field are required for dynamo (large scale) action? Theory of onset of dynamo action, but not for non-linear saturation More detailed discussion of induction equation 14 / 74 Ohm s law Ohm s law Equation of motion for drift velocity v d of electrons ( vd n e + v ) d = n e q e (E + v d B) p e τ ei τ ei : collision time between electrons and ions n e : electron density q e : electron charge : electron mass p e : electron pressure With the electric current: j = n e q e v d this gives the generalized Ohm s law: j + j τ ei = n eq 2 e E + q e j B q e p e Simplifications: τ ei ω L 1, ω L = eb/ : Larmor frequency neglect p e low frequencies (no plasma oscillations) 15 / 74 Simplified Ohm s law with the plasma conductivity j = σe σ = τ ein e q 2 e The Ohm s law we derived so far is only valid in the co-moving frame of the plasma. Under the assumption of non-relativistic motions this transforms in the laboratory frame to j = σ (E + v B) 16 / 74 Induction equation* Advection, diffusion, magnetic Reynolds number Using Ampere s law B = µ 0 j yields for the electric field in the laboratory frame leading to the induction equation with the magnetic diffusivity E = v B + 1 µ 0 σ B = E = (v B η B) η = 1 µ 0 σ. L: typical length scale U: typical velocity scale L/U: time unit ( = v B 1 ) B R m with the magnetic Reynolds number R m = U L η. 17 / / 74

4 Advection, diffusion, magnetic Reynolds number Advection, diffusion, magnetic Reynolds number R m 1: diffusion dominated regime = η B. Only decaying solutions with decay (diffusion) time scale τ d L2 η Object η[m 2 /s] L[m] U[m/s] R m τ d earth (outer core) years sun (plasma conductivity) years sun (turbulent conductivity) years liquid sodium lab experiment s R m 1 advection dominated regime (ideal MHD) Equivalent expression = (v B) = (v )B + (B )v B v advection of magnetic field amplification by shear (stretching of field lines) amplification through compression 19 / / 74 Advection, diffusion, magnetic Reynolds number Incompressible fluid ( v = 0): db = (B )v Velocity shear in the direction of B plays key role. Mathematically similar equation for compressible fluid (Walen equation): ( ) d B B ϱ = ϱ v Vertical flux transport in statified medium: B ϱ B ϱ 2/3 B ϱ 1/2 B = const. no expansion in direction of B isotropic expansion 2D expansion in plane containing B only expansion in direction of B Alfven s theorem Let Φ be the magnetic flux through a surface F with the property that its boundary F is moving with the fluid: Φ = B df dφ = 0 Flux is frozen into the fluid Field lines move with plasma F 21 / / 74 Dynamos: Motivation For v = 0 magnetic field decays on timescale τ d L 2 /η Earth and other planets: Evidence for magnetic field on earth for years while τ d 10 4 years Permanent rock magnetism not possible since T > T Curie and field highly variable field must be maintained by active process Sun and other stars: Evidence for solar magnetic field for years ( 10 Be) Most solar-like stars show magnetic activity independent of age Indirect evidence for stellar magnetic fields over life time of stars But τ d 10 9 years! Primordial field could have survived in radiative interior of sun, but convection zone has much shorter diffusion time scale 10 years (turbulent diffusivity) Mathematical definition of dynamo S bounded volume with the surface S, B maintained by currents contained within S, B r 3 asymptotically, = (v B η B) in S B = 0 outside S [B] = 0 across S B = 0 v = 0 outside S, n v = 0 on S and 1 E kin = 2 ϱv2 dv E max t S v is a dynamo if an initial condition B = B 0 exists so that E mag = 1 B 2 dv E min t 2µ 0 23 / / 74

5 Mathematical definition of dynamo Large scale/small scale dynamos Is this dynamo different from those found in powerplants? Decompose the magnetic field into large scale part and small scale part (energy carrying scale of turbulence) B = B + B0 : Z Z Emag = B dv + B dv. 2µ0 2µ0 Both have conducting material and relative motions (rotor/stator in powerplant vs. shear flows) Difference mostly in one detail: Dynamos in powerplants have wires (very inhomogeneous conductivity), i.e. the electric currents are strictly controlled Mathematically the system is formulated in terms of currents A short circuit is a major desaster! For astrophysical dynamos we consider homogeneous conductivity, i.e. current can flow anywhere Mathematically the system is formulated in terms of B (j is eliminated froquations whenever possible). A short circuit is the normal mode of operation! 2 Small scale dynamo: B B0 2 Large scale dynamo: B B 2 02 Almost all turbulent (chaotic) velocity fields are small scale dynamos for sufficiently large Rm, large scale dynamos require additional large scale symmetries (see second half of this lecture) Homogeneous vs. inhomogeneous dynamos 25 / 74 What means large/small in practice (Sun)? 26 / 74 What means large/small in practice (Sun)? Figure: Numerical sunspot simulation. Dimensions: Left 50x50 Mm, Right: 12.5x12.5 Mm Figure: Full disk magnetogram SDO/HMI 28 / / 74 Small scale dynamo action SSD in solar photosphere: kinematic phase Lagrangian particle paths: dx1 = v(x1, t) dx2 = v(x2, t) Consider small separations: δ = x1 x2 dδ = (δ )v Chaotic flows have exponentially growing solutions. Due to mathematical simularity the equation: d B B = v % % has exponentially growing solutions, too. We neglected here η, exponentially growing solutions require Rm > O(100). 29 / / 74

6 SSD in solar photosphere: saturated phase SSD in solar photosphere: power spectra Movie Movie Kinematic phase: Magnetic energy peaks at smallest resolved scales (here 30 km (4 km numerical resolution, would be m for the Sun Saturated phase: Magnetic energy peaks at granular scales (mostly flat spectrum at large scales). Dynamo action moved toward larger scales, where most of the kinetic energy sits (downflow lanes 300 km) 31 / / 74