Integrated electromotive response of a finite cyclonic event (${\mathcal{E}}_x=\pi B_0{a}^4(x\cos x-\sin x)/x^2$)

For the ideal magnetic response to a cylindrical cyclonic event, the plane-integrated $x$-component of $\mathbf{u}\times \mathbf{B}$ at duration $t$ isThis follows from $(\mathbf{u}\times \mathbf{B}{)}_x=(rf{B}_z+g{A}_r)\cos \varphi -(g/r)A_{\varphi }\sin \varphi$ and azimuthal integration. If boundary terms vanish, integration by parts gives the equivalent form $\pi B_0{\int }_0^{\infty }{r}^2{g}^{\prime }[\sin (ft)+tf\cos (ft)]dr$. For $f=g=a^2-r^2$ on $r<a$, zero outside, the response isThe value at zero is its continuous limit zero. It is initially $-\pi B_0{a}^{6}t/3$, but changes sign repeatedly at the nonzero roots of $\tan x=x$, with oscillatory envelope of order $t^{-1}$. The flow's kinetic helicity density is $2f^2\ge 0$, showing that the sign of a finite-duration ideal response is not fixed by the instantaneous helicity sign.