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Small- and Large-Scale Dynamo: Kinematic Theory Stanislav Boldyrev (Wisconsin-Madison) & Fausto Cattaneo (Chicago) Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas MHD Equations ∂τv + (v¢r)v = -rp + (r£B)£B + ν∆v + f Kinematic dynamo ∂τB = r£(v£B) + η∆B Re=VL/ν - Reynolds number Rm=VL/η - magnetic Reynolds number Pm=ν/η - magnetic Prandtl number B V1 B V2 Kinematic Turbulent Dynamo: Phenomenology V(x, t) is given. ∂τB = r£(v£B) + η∆B Consider turbulent velocity field V(x,t) with the spectrum: EK δVλ / λ1/3 K-5/3 λ» 1/K τλ» λ/δVλ/ λ-2/3 smaller eddies rotate faster K0 Kν K Magnetic field is most efficiently amplified by the smallest eddies in which it is frozen. The size of such eddies is defined by resistivity. Kinematic Turbulent Dynamo: Phenomenology EK Role of resistivity η K-5/3 EM(K) K0 Kη Kν Kη K Small Prandtl number, Large Prandtl number, PM=ν/η ¿ 1. PMÀ 1. Dynamo growth rate: Dynamo growth rate: γ» 1/τη γ » 1/τν Phenomenology: Large Prandtl Number Dynamo EK K-5/3 Large Prandtl number PM=ν/η À 1 EM(K) Interstellar and intergalactic media K0 Kν Kη K Magnetic lines are folded B λη Cattaneo (1996) Schekochihin et al (2004) Phenomenology: Large Prandtl Number Dynamo EK K-5/3 Large Prandtl number PM=ν/η À 1 EM(K) Interstellar and intergalactic media K0 Kν Kη K Folded fields in astrophysics Radio wave scattering in the Non-thermal radio filaments Galactic center is caused by in the GC may have “folded folded fields structure” Goldreich & Srihdar (2006) S. B. & Farhad Yusef-Zadeh (2006) Phenomenology: Small Prandtl Number Dynamo EK Small Prandtl number K-5/3 PM=ν/η ¿ 1 Stars, planets, liquid metal EM(K) experiments K0 Kη Kν K δVλ » λ1/3 τλ » λ/δVλ / λ2/3 γ λη / (RM)-3/4 Dynamo growth rate: γ » 1/τλη/ (RM)1/2 RM [S.B. & F. Cattaneo (2004)] Numerics: Haugen et al (2004) Kinematic Turbulent Dynamo: Theory ∂τB = r£(v£B) + η∆B V(x, t) is a given turbulent field Two Major Questions: 1. What is the dynamo threshold, i.e., the critical magnetic Reynolds number RM, crit ? 2. What is the spatial structure of the growing magnetic field (characteristic scale, spectrum)? These questions cannot be answered from phenomenological dimensional estimates! Kinematic Turbulent Dynamo: Theory Dynamo is a net effect of magnetic line stretching and resistive reconnection. RM > RM, crit : stretching wins, RM < RM, crit : reconnection wins, dynamo no dynamo RM=RM, crit: stretching balances reconnection: λη B When RM exceeds RM, crit only slightly, it takes many turnover times to amplify the field Kinematic Turbulent Dynamo: Kazantsev Model homogeneity and isotropy incompressibility No Dynamo Dynamo Kazantsev Model: Large Prandtl Number EK K-5/3 EM(K) If we know ψ(r, t), we know growth rate and spectrum of magnetic filed K0 Kν Kη K Large Prandtl number: PM=ν/η À1 Kazantsev model predicts: 1. Dynamo is possible; 2. EM(K)/ K3/2 Numerics verify both predictions: Schekochihin et al (2004) Kazantsev Model: Small Prandtl Number EK K-5/3 PM=ν/η ¿ 1 EM(K) K0 Kη Kν K Is turbulent dynamo possible? Batchelor (1950): “analogy of magnetic field and vorticity.” NO Kraichnan & Nagarajan (1967): “analogy with vorticity does not work.” MAYBE Vainshtein & Kichatinov (1986): YES Present day direct numerical simulations: “no dynamo action, maybe it does not exist” ? Schekochihin et al; Ponty et al; Mininni et al (2004;2005; 2006) Small Prandtl Number: Dynamo Is Possible EK EK K-5/3 K-5/3 EM(K) EM(K) K0 Kν KηK K0 Kη Kν K PM=ν/η À 1 PM=ν/η ¿ 1 Keep RM constant. Add small-scale eddies (increase Re). Kazantsev model: dynamo action is always possible, but for PM¿1, the critical magnetic Reynolds number (RM=LV/η) is very high. Kazantsev Model: Small Prandtl Number Theory L/λη S. B. & Cattaneo (2004) RM, crit PMÀ 1 PM¿ 1 Simulations P. Mininni et al (2004) May be crucial for laboratory dynamo, PM¿ 1 Re Kinematic Dynamo with Helicity It is natural to expect that turbulence can amplify magnetic field at K ¸ K0 K-5/3 EK K0 Kν Kη K Can turbulence amplify magnetic field at K ¿ K0 ? Yes, if velocity field “large-scale dynamo” has nonzero helicity Dynamo with Helicity: Kazantsev Model h=s v¢ (r £ v)d3 x ≠ 0 given energy helicity magnetic energy magnetic helicity need to find Equations for M(r, t) and F(r, t) were derived by Vainshtein and Kichatinov (1986) Dynamo with Helicity: Kazantsev Model Two equations for magnetic energy and magnetic helicity can be written in the quantum-mechanical “spinor” form: r r h=s v¢ (r £ v)d3 x = 0 h=s v¢ (r £ v)d3 x ≠ 0 S. B. , F. Cattaneo & R. Rosner (2004) Kazantsev model and α-model α - model assumes scale separation r Kazantsev model, NO scale separation The α - model approaches the exact solution only at r→ 1 Conclusions 1. In simple cases, kinematic turbulent dynamo is relatively well understood both phenomenologically and analytically. 2. Separation of small- and large-scale may be not a correct procedure. 3. Practically no analytic results for anisotropic and inhomogeneous cases that are relevant for astrophysics...

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posted: | 3/25/2011 |

language: | English |

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