  The impact of shoaling on linear water waves is well-known, but it has only
been recently found to significantly amplify both the intensity and frequency
of rogue waves in nonlinear irregular wave trains atop coastal shoals. At least
qualitatively, this effect has been partially attributed to the "rapid'" nature
of the shoaling process, i.e., shoaling occurs over a distance far shorter than
that required for waves to modulate themselves and adapt to the reduced water
depth. Through the development of a theoretical model and highly accurate
nonlinear simulations, we disentangle the respective effects of the slope
length and the slope gradient of a shoal and focus on the slope length to
investigate the rapidness of the shoaling process on the evolution of key
statistical and spectral sea-state parameters. Provided the shoal slope is 1/10
or steeper, our results indicate that the non-equilibrium dynamics is involved
even for rather short shoaling lengths and becomes dominant in the regime of
large lengths. When the non-equilibrium dynamics governs the wave evolution,
further extending the slope length no longer influences the statistical and
spectral measures. Thus, the shoaling effect on rogue waves is mainly driven by
the slope magnitude rather than the slope length. Moreover, the simulations
show that a higher cut-off frequency of the wave spectrum has a smaller impact
on wave statistics than expected for a flat bottom in deep water and that
insufficient attenuation of low-frequency waves at the downstream domain
boundary has notable influence on wave statistics atop the shoal.
