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A higher-order-decay result for the dynamo equation with an application to the toroidal velocity theorem

Published online by Cambridge University Press:  03 June 2015

Ralf Kaiser
Ralf Kaiser*
Affiliation:
Fakultät für Mathematik, Physik und Informatik, Universität Bayreuth, 95440 Bayreuth, Germany, (ralf.kaiser@uni-bayreuth.de)
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Abstract

In dynamo theory the distinction between decaying (in time) magnetic fields and those that do not is of crucial importance. Often decay is not manifest for the magnetic field itself but only for a single component or a scalar potential. Typically these auxiliary quantities satisfy evolution equations of the same type as the original induction equation. We prove here for these equations a theorem relating the decay of a solution to the decay of its higher derivatives. This result allows us to relate the decay of an auxiliary quantity to that of the magnetic field and, moreover, to relate integral decay to pointwise decay. As an application we strengthen the ‘toroidal velocity theorem’ in that we demonstrate pointwise decay of the magnetic field and electric current to zero under the conditions of this theorem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 3 , June 2015 , pp. 535 - 557
Copyright
Copyright © Royal Society of Edinburgh 2015 

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References

1 Backus, G.. A class of self-sustaining dissipative spherical dynamos. Annals Phys. 4 (1958), 372–447.Google Scholar
2 Bullard, E. C. and Gellman, H.. Homogeneous dynamos and terrestrial magnetism. Phil. Trans. R. Soc. Lond. A 247 (1954), 213–278.Google Scholar
3 Cowling, T. O.. The magnetic field of sunspots. Mon. Not. R. Astr. Soc. 94 (1934), 39–48.Google Scholar
4 Elsasser, W. M.. Induction effects in terrestrial magnetism. I. Theory. Phys. Rev. 69 (1946), 106–116.Google Scholar
5 Ivers, D. J. and James, R. W.. Antidynamo theorems for non-radial flows. Geophys. Astrophys. Fluid Dynam. 40 (1988), 147–163.Google Scholar
6 Kaiser, R.. The non-radial velocity theorem revisited. Geophys. Astrophys. Fluid Dynam. 101 (2007), 185–197.Google Scholar
7 Kaiser, R.. A toroidal magnetic field theorem. Commun. Math. Phys. 290 (2009), 633–649.Google Scholar
8 Kaiser, R.. Well-posedness of the kinematic dynamo problem. Math. Meth. Appl. Sci. 35 (2012), 1241–1255.Google Scholar
9 Kaiser, R. and Tilgner, A.. The axisymmetric antidynamo theorem revisited. SIAM J. Appl. Math. 74 (2014), 571–597.Google Scholar
10 Kaiser, R. and Uecker, H.. Well-posedness of some initial-boundary-value problems for dynamo-generated poloidal magnetic fields. Proc. R. Soc. Edinb. A 139 (2009), 1209–1235. (Corrigendum. Proc. R. Soc. Edinb. A 141 (2011), 819–824.)Google Scholar
11 Robinson, J. C.. Infinite-dimensional dynamical systems (Cambridge University Press, 2001).Google Scholar
12 Schmitt, B. J.. The poloidal–toroidal representation of solenoidal fields in spherical domains. Analysis 15 (1995), 257–277.Google Scholar
13 Stredulinsky, E. W., Meyer-Spasche, R. and Lortz, D.. Asymptotic behavior of solutions of certain parabolic problems with space and time dependent coefficients. Commun. Pure Appl. Math. 39 (1986), 233–266.Google Scholar