Abstract
Magnetic fields of planets and stars are believed to be generated and maintained by dynamo action, because the timescale of their dissipation is shorter than their lifetime. For dynamo processes to act, however, a seed field is required, which will be brought in from outside when the objects are formed or generated inside them. In this chapter, as preliminary studies of dynamo processes, Biermann’s battery mechanism, and Cowling’s anti-dynamo theorem are introduced. Then, the essences of the mean field dynamo by Steenbeck–Krause–Rädler and Parker’s cyclonic dynamo are described.
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- 1.
Chris A. Jones’s Les Houches Summer School 2007 Notes on Dynamo Theory. Available electrically at http://www.maths.leeds.ac.uk/cajones/LesHouches.html.
- 2.
A formula of vector analyses gives, for arbitrary u and B,
$$\displaystyle \begin{aligned}\mathrm{curl}({\boldsymbol u}\times{\boldsymbol B})= ({\boldsymbol B}\cdot\nabla){\boldsymbol u} -({\boldsymbol u}\cdot\nabla){\boldsymbol B}+{\boldsymbol u}\, \mathrm{div}\,{\boldsymbol B} -{\boldsymbol B}\, \mathrm{div}\,{\boldsymbol u}. \end{aligned}$$Furthermore, for arbitrary u and B we have
$$\displaystyle \begin{aligned}{}[({\boldsymbol B}\cdot\nabla){\boldsymbol u}]_\varphi=\bigg(B_r\frac{\partial}{\partial r}+ B_\varphi\frac{\partial}{r\partial \varphi} +B_z\frac{\partial}{\partial z}\bigg)u_\varphi +\frac{B_\varphi}{r}u_r. \end{aligned}$$For arbitrary vector A, ∇2 A is defined by
$$\displaystyle \begin{aligned}\nabla^2{\boldsymbol A}\equiv\mathrm{grad}(\mathrm{div}\,{\boldsymbol A})-\mathrm{curl}(\mathrm{curl}\, {\boldsymbol A}). \end{aligned}$$Also, for example, we have
$$\displaystyle \begin{aligned}(\mathrm{curl}\,{\boldsymbol A})_z=\frac{1}{r}\frac{\partial}{\partial r}(rA_\varphi)-\frac{\partial A_r}{r\partial\varphi}, \end{aligned}$$where A r and A φ are, respectively, r- and φ-components of A.
- 3.
It is noticed that B p is related to the r- and z-components of A φ.
- 4.
It is noted that the cyclonic dynamo which will be considered in Sect. 14.3.3 corresponds to the opposite case, i.e., \(\mathcal {R}_m\gg 1.\)
- 5.
The condition of neglecting the viscous term is ν∕l 2 < 1∕δt, which is \(\delta t<l^2/\nu =(l/u)\mathcal {R}\). The inequality (14.14) is satisfied if \(\mathcal {R}\) is not too small.
- 6.
The term of ρ u (1) can be derived from the zeroth-order quantities, since ∇2 p (1) can be expressed by the zeroth-order quantities by use of the assumption of incompressibility.
- 7.
ε ijk is the unit alternative tensor. When there is the same subscripts among i, j, k, we have ε ijk = 0. When i, j, and k are different, and they are the same cycle as 1, 2, 3, it is unity, while if others, it is − 1.
References
Biermann, L.: Zeit. Naturforsch. A5, 65 (1950)
Cowling, T.G.: Mon. Not. Roy. Astron. Soc. 94, 39 (1933)
Dolginov, A.Z.: Astr. Astrophys. 54, 17 (1977)
Mestel, L., Moss, D.L.: Mon. Not. R. Astron. Soc. 1983(204), 557 (1983)
Mestel, L., Roxburgh, I.W.: Astrophys. J. 136, 615 (1962)
Parker, E.N.: Astrophys. J. 162, 665 (1970)
Steenbeck, M., Krause, F., Rädler, K.-H.: Zeit. Naturforsch. A21, 369 (1966)
Yoshimura, H.: Astrophys. J. 201, 740 (1975)
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Kato, S., Fukue, J. (2020). Astrophysical Dynamo. In: Fundamentals of Astrophysical Fluid Dynamics. Astronomy and Astrophysics Library. Springer, Singapore. https://doi.org/10.1007/978-981-15-4174-2_14
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DOI: https://doi.org/10.1007/978-981-15-4174-2_14
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