Nonlinear evolution of viscous overstability in N-body simulations of a perturbed ring. Left panels: curves of kinetic energy density (Eq. (22)) of perturbations in units , excluding the ground state velocities (Eq. (53)). The curves are sampled at quadrature such that the radially averaged optical depth takes the value with time-averaged ground state optical depth τ₀ = 1.5. Right panels: snapshots of the radial surface density profile for each q taken at the same final time at quadrature where and the radial width of the simulation region takes the value . The profiles are scaled with the radially averaged surface mass density , which oscillates about its time-average σ₀ = 300 kg m⁻² in the same way as Eq. (69). Furthermore, a vertical frequency enhancement of Ω_z ∕Ω = 2 was used.