We prove that a Gaussian monopole, also known as the Lamb-Oseen vortex, is the only localized, rotating, self-similar solution to the two-dimensional, incompressible Navier-Stokes equations where level sets of vorticity and corotating streamfunction coincide. Our definition of self-similarity is restricted to the natural linear combination of space, time and viscous diffusion. We arrive at this conclusion by analytically determining the azimuthal Fourier modes for all possible solutions to this

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... problem and then proving that the amplitude of all but the first (axisymmetric) is zero. Since coherent vortex multipoles are observed to be in a state where lines of vorticity and corotating streamfunction correspond, this casts doubt on the existence of any self-similar asymptotic structure other than the monopole.