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On generalised toroidal-poloidal solutions of vector field equations

Published online by Cambridge University Press:  17 February 2009

D. J. Ivers
D. J. Ivers
Affiliation:
Department of Applied Mathematics, University of Sydney, N.S.W. 2006, Australia.
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Abstract

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The orthogonal coordinate systems ξi(i = 1, 2, 3) are determined, in which the gneralised toroidal and poloidal fields, defined respectively by T{T} = ∇ × {T∇ξ1} and S{S} = ∇ × T{S}, have the following three properties:

GP1 Decoupling of the vector Helmholtz equation: There exist linear differential operators L1 and L2 such that Hu = 0, where H is the vector Helmholtz operator [see equation (1)] and u = T{T} + S {S}, if and only if L1T = 0 and L2S = 0.

GP2 Orthogonality

GP3 Closure: ∇ × S{S} is a T field.

Two choices of T and S fields are considered: type I fields with potentials T and S, which may depend on ξ1, ξ2 and ξ3, and type II fields with ξ1-independent potentials. It is shown that properties GP1–GP3 only hold for type I fields in spherical and cylindrical coordinate systems, and for type II fields in azimuthal and cylindrical coordinate systems with axisymmetric and two-dimensional potentials, respectively. Analogues of GP1 for the vector wave and diffusion equations, and the Navier equation of linear elasticity, are also only true in the same four cases. Generalisations of type I and II T and S fields to arbitrary coordinate systems are indicated.

Type
Research Article
Information
The ANZIAM Journal , Volume 30 , Issue 4 , April 1989 , pp. 436 - 449
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Backus, G. E., “A class of self-sustaining dissipative spherical dynamos”, Ann. Phys, 4 (1958) 372–447.Google Scholar
[2]Bishop, R. L. and Goldberg, S. I., Tensor analysis of manifolds (Dover Publications, New York, 1980).Google Scholar
[3]Bullard, E. C. and Gellman, H., “Homogeneous dynamos and terrestrial magnetism”, Phil. Trans. Roy. Soc. Lond. A 247 (1954) 213–278.Google Scholar
[4]Chadwick, P. and Trowbridge, E. A., “Elastic wave fields generated by scalar wave functions”, Proc. Camb. Phil. Soc. 63 (1967) 1177–1187.Google Scholar
[5]Cowling, T. G., “The magnetic field of sunspots”, Mon. Not. Roy. Astron. Soc. 94 (1934) 39–48.Google Scholar
[6]Eringen, A. C. and Suhubi, E. S., Elastodynamics VOl. 2, Linear theory (Academic Press, New York, 1975).Google Scholar
[7]Forsyth, A. R., Differential geometry (Cambridge University Press, Cambridge, 1912).Google Scholar
[8]Ivers, D. J. and James, R. W., “Axisymmetric antidynamo theorems in compressible nonuniform conducting fluids’, Phil. Tans. Roy. Soc. Load. A 312 (1984) 179–218.Google Scholar
[9]Lamb, H., “On the oscillations of a viscous spheroid”, Proc. Lond. Math. Soc. 13 (1881), 51–66.Google Scholar
[10]Lortz, D., “Exsct solutions of the hydromagnetic dynamo problem”, Plasma Phys. 10 (1968) 967–972.Google Scholar
[11]Moon, P. and Spencer, D. E., Field theory for engineers (Van Nostrand, Princeton, 1961).CrossRefGoogle Scholar
[12]Morse, P. M. and Feshbach, H., Methods of theoretical physics, Part II (McGraw-Hill Book Company, New York, 1953).Google Scholar
[13]Rädler, K. H., “Zur dynamotheorie kosmischer magnetfelder. II. Darstellung von vektorfeldern als summe aus einem poloidalen und einem toroidalen anteil”, Astron. Nachr. 295(1974) 73–84.Google Scholar
[14]Vainshtein, S. I., “Simplest dynamo instability’, Soy. Phys. J.E.T.P. 41 (1976) 494–497.Google Scholar
[15]Willmore, T. J., An introduction to differential geometry (Oxford University Press, Oxford, 1959).Google Scholar