Safe zone for phase-resolved simulation of interactions between waves and vertically sheared currents

In this letter, we analyze the numerical stability of a velocity-based boundary integral equation for nonlinear interactions between surface waves and vertically sheared currents and propose an upper bound for the wave elevations associated with a finite number of resolved wave modes to guarantee the convergence of the numerical solution. The upper bound is expressed as a function of the L^∞ norm of dimensionless wave elevations and wave steepness. In general, energy accumulation at high wavenumbers may lead to numerical instability. The upper bound decreases as we increase the number of resolved wave modes. Furthermore, when we assume the regularity of the wave field, such as the power-law decay in the spectral domain, the upper bound becomes independent of the number of resolved wave modes when it becomes large enough, and the criteria of simulation stability are relaxed.