Flow and magnetic structures in a kinematic ABC-dynamo
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August 2020 Vol. 63 No. 8: 284712 https://doi.org/10.1007/s11433-019-1568-x
Flow and magnetic structures in a kinematic ABC-dynamo Tao Zhang1, ZhiLiang Lin1,2* , ChenYang Huang1, and Alistair G. L. Borthwick3 1 State
Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China; 2 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China; 3 School of Engineering, The University of Edinburgh, Edinburgh EH9 3JL, UK Received December 19, 2019; accepted April 23, 2020; published online June 22, 2020
Dynamo theory describes the magnetic field induced by the rotating, convecting and electrically conducting fluid in a celestial body. The classical ABC-flow model represents fast dynamo action, required to sustain such a magnetic field. In this letter, Lagrangian coherent structures (LCSs) in the ABC-flow are detected through Finite-time Lyapunov exponents (FTLE). The flow skeleton is identified by extracting intersections between repelling and attracting LCSs. For the case A = B = C = 1, the skeleton structures are made up from lines connecting two different types of stagnation points in the ABC-flow. The corresponding kinematic ABC-dynamo problem is solved using a spectral method, and the distribution of cigar-like magnetic structures visualized. Inherent links are found to exist between LCSs in the ABC-flow and induced magnetic structures, which provides insight into the mechanism behind the ABC-dynamo. kinematic dynamo theory, ABC-flow, Lagrangian coherent structure PACS number(s): Citation:
91.25.Cw, 91.25.Mf, 47.27.De
T. Zhang, Z. L. Lin, C. Y. Huang, and A. G. L. Borthwick, Flow and magnetic structures in a kinematic ABC-dynamo, Sci. China-Phys. Mech. Astron. 63, 284712 (2020), https://doi.org/10.1007/s11433-019-1568-x
1 Introduction In 1965, Arnold et al. [1, 2] discovered a class of analytical solutions of the Euler equations whose flow exhibited “fast dynamo” action. This type of flow was later named ArnoldBeltrami-Childress (ABC) flow, and may be expressed as: x˙ = A sin z + C cos y, y˙ = B sin x + A cos z, z˙ = C sin y + B cos x,
(1)
where x, y and z are tracer locations in cartesian space, A, B and C are three prescribed parameters that control the dynamo action of the flow. In 1986, Dombre et al. [3] identified *Corresponding author (email: [email protected])
a relationship between the three parameters and the positions of different categories of flow stagnation points. In the same year, Galloway et al. [4] investigated the properties of a fast dynamo flow by varying the magnetic Reynolds number with prescribed values of the parameters A, B, and C, and discovered the presence of cigar-like structures in the magnetic field. In 2003, Archontis et al. [5, 6] separately combined the magnetic field with laminar and turbulent flow fields, and found that the difference in growth rate is due to discrepancies in the recycling of the weake
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