Long-time behavior of three-dimensional gravity-capillary solitary waves on deep water generated by a moving air-blowing forcing: Numerical study

Abstract

Long-time simulations are conducted on a forced three-dimensional (3D) nonlinear viscous gravity-capillary wave equation that describes the surface wave pattern when the forcing moves on the surface of deep water with speeds less than the linear phase speed c(min) = 23 cm/s. Three different states are identified according to forcing speeds U below c(min). At relatively low speeds below a certain speed (c(1)), a steady circular dimple is observed below the moving forcing. At relatively high speeds above a certain speed (c(2)), "symmetric" shedding phenomena of 3D depressions are observed behind the moving forcing. At intermediate speeds (c(1)<= U <= c(2)), steady 3D gravity-capillary solitary waves are generated behind the moving forcing and are maintained for some time. After long-time simulations, however, those gravity-capillary solitary waves break up and 3D local depressions are shed asymmetrically behind the moving forcing. In more detail, when the forcing speed (U) is very close to c(1), the asymmetric shedding is "almost regular" and when the forcing speed (U) is very close to c(2), the asymmetric shedding is "regular antisymmetric," after a transient period of an "irregular" asymmetric shedding from the steady state of 3D gravity-capillary solitary waves. On the contrary, for the remaining cases of the entire forcing speeds (c(1) < U < c(2)), the asymmetric shedding is "irregular.