The Stability of a “uniform” Amplitude Wavetrain

A narrow-banded spectrum of gravity waves propagating in one horizontal direction 4 at the surface of water with finite depth may be governed by a nonlinear Schrödinger equation 5 (NLSE). If the depth is non-uniform, the coefficients in the NLSE are variable and an additional 6 linear term due to conservation of wave action flux is present, which may act as a dissipative term 7 that causes the wave envelope to broaden and decrease in amplitude or which may act as a growth 8 term that causes the wave envelope to focus and increase in amplitude. The addition of linear, viscous 9 dissipation either enhances the dissipative process or competes with the growth effect. Here we 10 consider such a variable-coefficient, dissipative NLSE. We find a solution that has uniform amplitude 11 in the appropriate reference frame and examine its stability. We find that for waves propagating 12 into shallower water, both bathymetry and viscous dissipation act to stabilize the modulational 13 instability. For waves propagating into deeper water, bathymetry acts to enhance instability, but 14 viscous dissipation eventually causes stabilization. For a special bathymetry, these two effects balance 15 so that the waves are governed by an inviscid NLSE and are consequently unstable to modulational 16 perturbations; this situation is considered for waves on oceanographic and laboratory scales. 17