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Stability of Long Helical Wave Length Free Boundary Equilibria With Slightly Ell

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Stability of Long Helical Wave Length Free Boundary Equilibria With Slightly Ell
H Weitzner
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Terms. Clearly, if we set A = IB, ^ will represent the m = -2 mode. G. N=2orm=+3 We again take ^ of order c, but now we must compute ^ 2 correct to order c, since there may be an admixture of m = + 1 in the higher order terms. To leading order in c, we again take over the old result in our new notation £ /, / 2 2, , ^y- V -ikz', 2£ 1^-ikz' ^^ = ('Alx -y ) +2lxy)e + ^ ^x ® ly = (-2^xy+, X(x2-y2))e-^^^' +^ ^ ® "^ ' so that fi = fi^ +c^a^ = ((k^ +4)^- 6ikJC)(^-xy^) + (6ik^+ (k^ +4)^) (x^y-y3/3) +c...^a^ we find -Q = -a +c(0, 0 +0 0, )e° - 2c0, e° +2cl° - cD°?J, x o, x ^lo ol^x ly X XX (C) o 1 -n = -Q +c(0, 0 +0 0, )? +2c0, ?"-2c^ - cD"^;^ ^, y o, y ^lo ol^y ix y yy' where D°, D° are the leading order differential operators. We readily see -14- 0^(x2-y2) ^ Q 2 2 0-^ xy = X +y We shall see that the right hand side of (C) is a gradient with ^-^ = if ^° corresponds to a m = -3 mode with k = 1 +0(c) and thus introduction of ? corresponds to a redefinition of the constants ^, X' A. Direct evaluation shows {\ A.

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