Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Optimal energy bounds in spherically symmetric $ \alpha^2$-dynamos


Authors: Ralf Kaiser and Andreas Tilgner
Journal: Quart. Appl. Math.
MSC (2010): Primary 49R05; Secondary 76W05, 85A30
DOI: https://doi.org/10.1090/qam/1501
Published electronically: February 21, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In kinematic dynamo theory energy bounds provide necessary conditions for dynamo action valid for every velocity field. When expressed by the magnetic Reynolds number $ R$ this number $ R_E$ may be compared with the critical Reynolds number $ R_c = R_c (\mathbf {v})$ indicating the onset of dynamo action for a given velocity field $ \mathbf {v}$. Typically, there is an (often large) gap between both numbers, which suggests the question: are there better (energy) bounds or are the most critical velocity fields not yet known (or are both conjectures false)?

Here we answer this question in a simplified setting, viz. for spherically symmetric $ \alpha ^2$-mean-field dynamos, where the single scalar field $ \alpha $ takes the role of the velocity field and where spherical symmetry allows the reliable numerical solution of a non-linear variational problem. The non-linear problem arises from the simultaneous variation of magnetic field and $ \alpha $-profile (measured in a suitable norm), which, in fact, yields an improved energy bound $ R_E^{opt} = 4.4717$ compared to the best hitherto known bound $ R_E = 3.0596$. This bound is close to the best hitherto known critical Reynolds number $ R_c = 4.4934$, which belongs to a constant $ \alpha $-profile, and is, moreover, optimal since it is connected to an $ \alpha $-profile whose critical Reynolds number exceeds $ R_E^{opt}$ by less than $ 10^{-4}$.


References [Enhancements On Off] (What's this?)

  • [Ada75] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [Bac58] George Backus, A class of self-sustaining dissipative spherical dynamos, Ann. Physics 4 (1958), 372-447. MR 0095004
  • [CHJ15] L. Chen, W. Herreman, and A. Jackson, Optimal dynamo action by steady flows confined to a cube, J. Fluid Mech. 783 (2015), 23-45. MR 3442467
  • [CH53] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • [HLP88] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Reprint of the 1952 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. MR 944909
  • [Hol03] R. Holme, Optimized axially-symmetric kinematic dynamos, Phys. Earth Planet. Inter. 140, 3-11 (2003).
  • [KBCWF12] I. V. Khalzov, B. P. Brown, C. M. Cooper, D. B. Weinberg, and C. B. Forest, Optimized boundary driven flows for dynamos in a sphere, Physics of Plasmas 19, 112106 (2012).
  • [Kai12] Ralf Kaiser, Well-posedness of the kinematic dynamo problem, Math. Methods Appl. Sci. 35 (2012), no. 11, 1241-1255. MR 2945849
  • [KU09] R. Kaiser and H. Uecker, Well-posedness of some initial-boundary-value problems for dynamo-generated poloidal magnetic fields, Proc. R. Soc. Edinburgh 139A, 1209-1235 (2009), Corrigendum, Proc. R. Soc. Edinburgh 141A, 819-824 (2011), Corrected version, arXiv:1212.3180 [astro-ph.SR] (2012). MR 2557319; MR 2819713
  • [KR80] F. Krause and K.-H. Rädler, Mean-field magnetohydrodynamics and dynamo theory, Pergamon Press, Oxford-Elmsford, N.Y., 1980. MR 635447
  • [LG96] J. J. Love and D. Gubbins, Optimized kinematic dynamos, Geophys. J. Int. 124, 787-800 (1996).
  • [Pro77] M. R. E. Proctor, On Backus' necessary condition for dynamo action in a conducting sphere, Geophys. Astrophys. Fluid Dynam. 9, 89-93 (1977).
  • [Pro07] M. R. E. Proctor, Homogeneous dynamos, Mathematical Aspects of Natural Dynamos, Ed. E. Dormy & A.M. Soward (Chapman & Hall/CRC, Boca Raton, USA 2007), pp. 18-41.
  • [SGG99] F. Stefani, G. Gerbeth, and A. Gailitis, Velocity profile optimization for the Riga dynamo experiment, Transfer Pheneomena in Magnetohydrodynamic and Electroconducting Flows, Springer, 1999, pp. 31-44.
  • [Til97] A. Tilgner, A kinematic dynamo with a small scale velocity field, Phys. Lett. A 226, 75-79 (1997).
  • [Wil12] A. P. Willis, Optimization of the Magnetic Dynamo, Phys. Rev. Lett. 109, 251101 (2012).

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 49R05, 76W05, 85A30

Retrieve articles in all journals with MSC (2010): 49R05, 76W05, 85A30


Additional Information

Ralf Kaiser
Affiliation: Fakultät für Mathematik und Physik, Universität Bayreuth, D-95440 Bayreuth, Germany
Email: ralf.kaiser@uni-bayreuth.de

Andreas Tilgner
Affiliation: Institut für Geophysik, Universität Göttingen, D-37077 Göttingen, Germany
Email: andreas.tilgner@geo.physik.uni-goettingen.de

DOI: https://doi.org/10.1090/qam/1501
Keywords: Dynamo theory, $\alpha^2$-dynamo, energy bound, variational method
Received by editor(s): July 26, 2017
Published electronically: February 21, 2018
Article copyright: © Copyright 2018 Brown University

American Mathematical Society