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Magnetic dynamos in accretion disks Magnetic helicity and the theory of astrophysical dynamos Dmitry Shapovalov JHU, 2006 Outline - Turbulence and magnetic fields in astrophysics - Dynamo problem - How the dynamo works: - old theory: mean-field dynamo - new theory: role of magnetic helicity - How can we learn is it true? - Results: what we did so far - Future: what else can be done Cosmic magnetic fields Crucial: stellar and solar activity, star formation, pulsars, accretion disks, formation and stability of jets, cosmic rays, gamma-ray bursts Probably crucial: protoplanetary disks, planetary nebulae, molecular clouds, supernova remnants Role is unclear: stellar evolution, galaxy evolution, structure formation in the early Universe Probably unimportant: planetary evolution Cosmic magnetic fields - Polarization of radiation: orientation || B Orion (Zweibel & Heiles, 1997, Nature, 385, 131) Cosmic magnetic fields - Polarization of radiation: orientation || - Zeeman splitting: B B Cosmic magnetic fields - Polarization of radiation: orientation || - Zeeman splitting: B B - Synchrotron radiation: intensity B2 / 7, polarization B NGC 2997 Cosmic magnetic fields - Polarization of radiation: orientation || - Zeeman splitting: B B - Synchrotron radiation: intensity B2 / 7, polarized B - Faraday rotation: for lin. polarized waves: RM Han et al., 1997, A&A 322, 98 ne B dl Cosmic magnetic fields - Polarization of radiation: orientation || - Zeeman splitting: B B - Synchrotron radiation: intensity B2 / 7, polarized B - Faraday rotation: for lin. polarized waves: RM ne B dl - direct measurements for the Sun, solar wind & planets TRACE satellite, 1998-2006 Vieser & Hensler, 2002 Turbulence Observed / predicted in: - convective zones of the stars and planets - stellar wind and supernova explosions - interstellar medium, both neutral and ionized - star forming regions - accretion disks - motion of galaxies through IGM Freyer & Hensler, 2002 Vieser & Hensler, 2002 Turbulence Observed / predicted in: - convective zones of the stars and planets - stellar wind and supernova explosions - interstellar medium, both neutral and ionized - star forming regions - accretion disks - motion of galaxies through IGM Freyer & Hensler, 2002 Origin of the cosmic magnetic fields 1. Origin of the weak initial local or uniform “seed” field 2. Amplification of the seed field Origin of the cosmic magnetic fields 1. Origin of the weak initial local or uniform “seed” field 2. Amplification of the seed field Theories range from fluctuations of hypermagnetic fields during the time of decoupling of electroweak interations (come together with baryonic asymmetry of the Universe), to various “battery effects”, which produce macroscopic seed fields on a continuing basic up to our time. One example is a Poynting-Robertson effect: M.Harwit, Astrophysical Concepts Origin of the cosmic magnetic fields 1. Origin of the weak initial local or uniform “seed” field 2. Amplification of the seed field Origin of the cosmic magnetic fields 1. Origin of the weak initial local or uniform “seed” field 2. Amplification of the seed field local (chaotic) seed field Small-scale dynamo strong intermittent field, with a scale of the largest eddies Origin of the cosmic magnetic fields 1. Origin of the weak initial local or uniform “seed” field 2. Amplification of the seed field local (chaotic) seed field uniform (large-scale) seed field Small-scale dynamo Large-scale dynamo strong intermittent field, with a scale of the largest eddies strong field with a largest scale available, EM EK Systems with large-scale dynamos - Earth, Jupiter, some other planets and their satellites - the Sun - accretion disks - some spiral galaxies - giant molecular clouds Earth - turbulence is driven by temperature gradient in liquid outer core - large-scale shear is given by Earth rotation - Re ~ 108, Rm ~ 350, resistive timescale ~ 2 105 years - B ~ 3 gauss (at CMB), exists for billions of years Radial component of the Earth’s field at core-mantle boundary (CMB) G.Rüdiger, The Magnetic Universe, 2004 Accretion disks - protostellar disks - close binaries - active galactic nuclei (AGNs) T Tauri YSO, image by NASA Illustration, D.Darling Accretion disks - protostellar disks - close binaries - active galactic nuclei (AGNs) Illustration, NASA Accretion disks - protostellar disks - close binaries - active galactic nuclei (AGNs) Quasar PKS 1127-145, image by Chandra Illustration, NASA/ M.Weiss Accretion disks - large-scale shear is given by Keplerian motion, - angular momentum L r1/ 2, i.e. it should be removed in some way when r 0 - ordinary viscosity is too small - turbulent viscosity requires turbulence and even then it will be small r 3/ 2 Accretion disks - large-scale shear is given by Keplerian motion, - angular momentum L r1/ 2, i.e. it should be removed in some way when r 0 - ordinary viscosity is too small - turbulent viscosity requires turbulence and even then it will be small - large-scale poloidal magnetic field can remove angular momentum from the system: r 3/ 2 Accretion disks - large-scale shear is given by Keplerian motion - even if Keplerian flow is stable over radial perturbations, in presence of vertical magnetic field turbulence can exist via MRI (magnetorotational instability, Balbus & Hawley, 1991) - MRI can drive the growth of azimuthal field, i.e. large-scale seed for a dynamo process Now we should explain how dynamo works Dynamo theory Dynamo theory Macroscopic magnetohydrodynamic (MHD) framework: V (V )V ( B) B V P f t B (V B) B t B 0, V 0 V velocity, B Alfven velocity B / 4 scale >> m.f.path, plasma scales (Larmor & Debye radii) velocity << sound speed (for incompressibility) Induction equation: B (V B ) B t 8 20 Solar flares: Rm 10 ; Galaxies: Rm 10 0 => magnetic fields are “frozen” into liquid, B can’t change its topology from small scales to the large ones => for dynamos 0 Mean-field electrodynamics (Moffatt, 78; Parker, 79) In differentially rotating object: radial seed field is stretched along the direction of rotation => azimuthal field grows To keep azimuthal field growing one needs to maintain radial component in some way: -effect <.> - “large-scale part” - “turbulent e.m.f.” t B ( v B ) 2 B , vb , t B tM j tV ; t B t j t H K 3, H K v dV , tM 2 E K 3; (from D.Biskamp, MHD Turbulence, 2003) H K ijk vi j vk - kinetic helicity: - symmetry should be broken in all 3 directions for H K 0 - doesn’t depend on any magnetic quantitities (, B) - “small-scale quantity”: direct cascade - not a conserved quantity in MHD, even for negligibly small - can’t support dynamo for a long time: magnetic back-reaction cancels all kinetic helicity at large scales (there is no preferred orientation for spirals) - -effect contradicts to simulations (Hughes & Cattaneo, 96, Brandenburg, 01) Mean-field theory have to be revised Magnetic helicity H M H A B, B A, M H alt ( A A0 ) ( B B0 )dV , A 0 B0 A0 Magnetic helicity H M H A B, - B A, A 0 HdV is conserved quantity in MHD: t H HV B A V J B, V B J H HV B A V - helicity current - H is the only integral in 3D, which has inverse cascade: can’t dissipate at small scales, remains at large ones, where resistivity is negligible, i.e. exists for a time bigger than dissipative timescale Htotal H h A B a b t H 2 B v b B A v b t h 2 B v b J h where J h ( a B )v b B ( a v ) v b , v B 2B v b - transfer of magnetic helicity between scales vb - turbulent e.m.f. (In mean-field treory t B Mean-field dynamo depends on the transfer of magnetic helicity between scales v b ) Simulations General features: - incompressible 3D MHD - pseudospectral (E, H - conserved, unlike in spatial code) - periodic box (H is gauge invariant) - resolution: from 64^3 to 1024^3 - timescale up to 100 eddy turnover times - both OpenMP & MPI parallel versions available Simulations Dynamo-specific features: - = (to simplify) - turbulence is driven by external random (gaussian) forcing - forcing has N components with variable spectral properties - forcing correlation time is variable - forcing has both linearly and circularly polarized components (for helicity injection into the turbulence) - divF =0 Simulations Dynamo-specific features: - turbulence is driven by external random (gaussian) forcing - forcing has N components with variable spectral properties - forcing correlation time is variable - forcing has both linearly and circularly polarized components (for helicity injection into the turbulence) - divF =0 - forcing is usually set at some fixed small scale (to simulate real systems) Simulations Dynamo-specific features: - forcing correlation time is variable - forcing has both linearly and circularly polarized components (for helicity injection into the turbulence) - divF =0 - forcing is usually set at some fixed small scale (to simulate real systems) - initial large scale shear and weak seed field: V , B V0 , B0 (eiky , 0, 0), k 1 Simulations Dynamo-specific features: - forcing has both linearly and circularly polarized components (for helicity injection into the turbulence) - divF =0 - forcing is usually set at some fixed small scale (to simulate real systems) - initial large scale shear and weak seed field (|Bo| << |Vo|): V , B V0 , B0 (eiky , 0, 0), k 1 - l.-s. shear is maintained const for anisotropy / against dissipative decay Results Energy evolution sm.scale forcing: kx/k = 1 Bo 106 Vo 0.7 Ro ~ 1, timespan ~ 10 e.t. _____________ small scale shear Ro large scale shear Ro kv Magnetic energy spectra Energy spectra sm.scale forcing: kx/k = 1 Bo 106 Vo 0.7 |k| time Ro ~ 1, Kinetic energy spectra timespan ~ 10 e.t. time |k| Magnetic field spectra Bx By time sm.scale forcing: kx/k = 1 |k| Bo 106 Vo 0.7 Ro ~ 1, Bz timespan ~ 10 e.t. B total Magnetic field spectra Bx By time sm.scale forcing: kx/k = 1 |k| | Bo | ~ | Vo |, Ro ~ 10, Bz timespan ~ 10 e.t. B total Evolution of large-scale magnetic energy for different initial large-scale fields Bo = 0.1 Bo = 0.01 Bo = 0.001 timesteps, 5K ~ 1.e.t. Evolution of large-scale magnetic energy for B 10 6 0 timesteps, 5K ~ 1.e.t. Magnetic helicity spectra time |k| All results we obtained so far already fit well into the helicity-based picture of the large-scale dynamo Future - next big goal is to prove numerically that turbulent e.m.f. depends on transfer of magnetic helicity between scales (balance formula) - then it is interesting to see how different terms in helicity current influence the dynamo - code: do subgrid modelling in order to expand dynamic range even more (to have everything covered: from largescale shear to the dissipation scale ) The end Magnetic energy spectra Energy spectra for different forcing spectral distribution Here: ky/k = 0.1 other parameters - “real-life” _______________________ time |k| Kinetic energy spectra No dynamo action when kx/k = 0 time |k| V, B = large scale v, b = small scale Vo, Bo - initial largescale fields ______________ Bx By time |k| Vo = Vx ~ exp(iky) Bo = Bx ~ exp(iky) sm.scale forcing: kx/k = 1 Bo 106 | Bo | << | Vo |, Ro ~ 10 50 Bz B total Balance formula t h 2 v b B jh BT B b, VT V v , ... jh (a B )v b1 a v B hV b( 2 a V ) h a b , 21 (v B ), 22 (V b) jh (a B)v b1 a v B hV b( 2 a V ) Simulations with real-life parameters • Ro ~ 1 in accretion disks, Sun’s convection zone. • Constant large-scale shear (to compensate disipative decay and to maintain non-uniformity in the system). • Weak initial magnetic field: |Bo| << |Vo|. We want large scale shear to help in generation of magnetic field from some small seed field. • Forcing correlation time ~ eddy turnover time. Small scale turbulence is driven by some instability which saturates when its growth time ~ eddy turnover time.