= Magnetic stabilization of an equal-density vortex sheet
{title2=$\sigma=-k_x\overline U\pm k_x\sqrt{v_A^2-(\Delta U)^2/4}$}
For two incompressible streams of equal density with a common tangential <magnetic field> parallel to their velocities, decaying <normal modes> and interface <displacement>/total-pressure continuity give $\rho(\sigma+k_xU_1)^2+\rho(\sigma+k_xU_2)^2=2k_x^2B^2$ in units $\mu_0=1$. The displayed <dispersion relation> follows, with $v_A^2=B^2/\rho$. Thus $|\Delta U|<2v_A$ excludes exponential <Kelvin-Helmholtz instability>; equality is marginal. Above threshold the <growth rate> is $|k_x|\sqrt{(\Delta U)^2/4-v_A^2}$. The transverse horizontal <wavenumber> affects the decay length through $k=(k_x^2+k_y^2)^{1/2}$ but not this growth threshold. <Magnetic tension> supplies the stabilizing restoring force.
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