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Forced solitary waves and hydraulic falls in two-layer flows

Long nonlinear waves in the two-layer flow of an inviscid, incompressible fluid are considered. Both the free surface and the interface of the two fluids are unknown free boundaries. The flow is forced by an obstruction on the bottom and/or an external pressure (usually called the wind stress) on the free surface. Away from the critical depth ratio between the surface layer and the internal layer of the fluids, the weak nonlinearity is of second order. Then there exists a balance between the dispersion and the second-order nonlinearity. The first-order asymptotic approximations of both free-surface elevation and interface elevation satisfy forced Korteweg-de Vries equations (fKdV). Because of the existence of two modes, the total number of types of solutions is double that for the single-layer flow. There are two hydraulic falls and four solitary waves for positive forcing. The first hydraulic fall and the first two solitary waves correspond to the fast mode. The remaining ones correspond to the slow mode. The first hydraulic fall was numerically found by Forbes (1989) by directly integrating the Laplace equation with nonlinear boundary conditions. There are two types of forcing according to the length of the base of the obstruction. One type is called local and the length of its base has the same scale as the height of the forcing. The expression of the forcing in the fKdV is given by the Dirac delta function. Based upon our long-wave scale the horizontal laboratory lengthscale is shrunk by ε½, while the vertical laboratory lengthscale is unchanged. Hence the semicircular bumps in the papers by Vanden-Broeck (1987), and Forbes (1988, 1989) are all considered to be local in the present paper. For the locally forced cases, stationary problems have been solved analytically. We have derived analytical expressions for the upstream speeds UL and Uc, at which the hydraulic falls can occur and solitary waves respectively cease to exist. The formulae are reduced to those due to Miles (1986) in the case of a single-layer flow and very small forcing. A full comparison among the asymptotic, computational and experimental results is provided. The comparison shows that the difference is less than 10% for UL and Uc in most of the parameter range where the asymptotic method is valid, the computational scheme converges and experiments were conducted (see figure 10). The second type of forcing is called non-local and the length of the base of the obstruction has the same scale as the wavelength. The existence and behaviour of the stationary solutions of the non-locally forced fKdV are described. Surprisingly, there can be more than four solitary waves sustained on the site of some negative forcing.