= Alfvén-resonant vorticity in a parallel magnetic shear flow
{c}
{title2=$[\rho S^2-k_x^2B^2](\zeta-k_yU'w/S)=0$}
For an incompressible <shear flow> $U(z)\mathbf e_x$ with uniform <magnetic field> $B\mathbf e_x$, define $S=\sigma+k_xU$ for time dependence $e^{i\sigma t}$ and $\zeta=ik_xv-ik_yu$. Eliminating the magnetic perturbations from the horizontal <momentum> equations gives the displayed product for $S\ne0$. Away from the <Alfvén wave> resonance, it yields $\zeta=k_yU'w/S$. At resonance one must retain the original system: for constant $U$, $k_y=0$, $S^2=k_x^2B^2/\rho$, any smooth transverse amplitude $v(z)$ with $b_y=k_xBv/S$ and all other amplitudes zero is an <Alfvén wave>. Its nonzero <vorticity> disproves the unrestricted zero-vorticity reduction. Exponentially growing modes with real background coefficients cannot encounter these real-frequency resonances.
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