Document Sample

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...

DOCUMENT INFO

Shared By:

Categories:

Tags:
Large Scale, small scale, Trading Company, Alibaba Group, Latin America, sensor networks, Test Report, spatial scales, long term, Food Scale

Stats:

views: | 9 |

posted: | 3/25/2011 |

language: | English |

pages: | 19 |

OTHER DOCS BY nikeborome

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.