By P. H Roberts
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Additional info for An introduction to magnetohydrodynamics
Weiss for permission to reproduce these results. They have recently appeared in print (Weiss, 1966). ;--~--~- - - - - . 4] Electromagnetic' Induction t=o·o t=0·5· t=l·O t=I·5 t=2·0 t=2·5 t =3,0 t=3·5 t=4·0 Fig. 5. The severing and coalescence of flux tubes. (Figure based on computer results obtained by N. O. s; I shown is u= (BlJI/By, -BlJI/Bx, 0) where 1 n The initial field (t = 0) is uniform, and subsequent diagrams show the lines of force at later times. The magnetic flux trapped between each pair of lines is the same in each picture.
2 (b) Fig. 3. Elementary volumes and circuits: (a) The penny-shaped disk (b) The rectangular circuit The dimensions of these are small compared with the radii of curvature of the surface and with the length scales of the fields considered. g. g. M. 6) , . I I t 1 ~--------~--------- The Equations of Magnetohydrodynamics be integrated round a contour crossing S [Fig. 3(b)] and Stokes' theorem applied. g. ). 2 are (0 x B) = 0, [from (5) and (9)], (81) (n x E) = 0, [from (6)], (82) (n. B) = 0, [from (7)], (83) [from (8) and (9)], [from (12)].
This is physically unreasonable since, by supposition, the only sources of energy lie in z < 0. J(ro/2 Q) exp i[Z (;y-wt]. J(ro/2 Q)exp i [Z (~) t - wt], o}. (22) On examining (21) and (22), we see that for any fixed t the directions of both Band j alternate periodically in z, with wavelength 2n (2Y/fw)t. J-. 2) increasing Z, the distance in which the amplitudes are reduced by a factor of l/e being (211/W)+. , (23) and 3, the time-scale characteristic of the disturbance, is the period of oscillation 2n/w.
An introduction to magnetohydrodynamics by P. H Roberts