Modulational stability of finite-depth Stokes waves with constant vorticity

We investigate the effect of constant vorticity on the modulational spectral stability of small-amplitude periodic gravity waves for the two-dimensional finite-depth water-wave system. We derive an explicit stability criterion that agrees with the prediction obtained by Thomas, Kharif and Manna [43] through a nonlinear Schrödinger approximation. For waves propagating to the right, we identify a sharp range of dimensionless positive (counterclockwise) vorticity, for which modulational instability is suppressed at every depth, while at any fixed depth sufficiently strong negative (clockwise) vorticity also restores modulational stability. We give a complete description of the spectrum near the origin and identify a figure-eight spectral curve in the unstable regime. A key distinction from the irrotational case is that the usual Hamiltonian factorization of the linearized operator fails at zero Floquet exponent on the full periodic Sobolev space. We therefore carry out a spectral reduction based on reversibility, without relying on Hamiltonian factorization.