Linear Instability of Symmetric Logarithmic Spiral Vortex Sheets

We consider Alexander spirals with \(M\ge 3\) $M\ge 3$ branches, that is symmetric logarithmic spiral vortex sheets. We show that such vortex sheets are linearly unstable in the \(L^\infty \) $L^{\infty }$ (Kelvin–Helmholtz) sense, as solutions to the Birkhoff–Rott equation. To this end we consider Fourier modes in a logarithmic variable to identify unstable solutions with polynomial growth in time.