Physics
Scientific paper
Jul 1970
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1970rspta.266..535r&link_type=abstract
Philosophical Transactions for the Royal Society of London. Series A, Mathematical and Physical Sciences, Volume 266, Issue 1179
Physics
46
Scientific paper
It is established analytically that, in a precisely defined sense, almost all steady spatially periodic motions of a homogeneous conducting fluid will give dynamo action at almost all values of the conductivity. The same result is obtained for motions periodic in space-time. The asymptotic form of the growing field, for an arbitrary initial field of finite energy, is also presented. Dynamo action is first shown to require that for some real vector j there is a magnetic field solution of the form B = H exp (pt + ij.x), where H is a complex function of position (or of position and time) with the same periodicity as the motion, and p has positive real part, indicating growth. This number p is an eigenvalue of a linear differential operator on the space of admissible functions H. The first term of a power series in j for the eigenvalues p which vanish to zero order is studied. It is thus proved sufficient for dynamo action that the determinant of the symmetric part of a certain 3 × 3 tensor, a function of the motion and conductivity, is non-zero. Finally, it is shown that this determinant is an analytic function of the conductivity, and is non-zero in a small conductivity limit for nearly all motions. This proves the stated result.
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