Nonlinear Gravity Waves in the Water Stream in the Channel With Complex Geometry of the Bottom Covered by Broken Ice

The two-dimensional nonlinear problem of a steady flow in a channel covered by broken ice with an arbitrary bottom topography is considered. Anintegral hodograph method is employed for solving the problem, which is reduced to a system of nonlinear equations in the velocity modulus on thefree surface. These equations are obtained from the dynamic boundary condition. Results showing the effect of gravity on the geometry of the free surfaceare presented for a wide rectangle and a trench over a wide range of Froude numbers, including both subcritical and supercritical flows. For supercriticalflows, two families of solutions for an arbitrary bottom shape are reproduced. It is shown that the additional condition requiring the free surface to be flat at a finite distance from the obstruction selects a unique solution for a given bottom height and width of the rectangle for supercritical flows. This solution is continuous in going from the subcritical to the supercritical flow regime. Case studies are carried out for a wide rectangular obstruction and a trench on the bottom of the channel. Two flow regimes are studied. The first is a subcritical regime, for which the wave generated by the bottom profile extends only downstream to infinity. The second is supercritical regime, for which two distinct types of solutions may exist: one with a smaller wave crest height, known as the ‘perturbed’ wave, and another with a larger wave crest height, termed the ‘soliton’ wave. The ‘perturbed’ wave belongs to a family of steady solutions that bifurcate from a uniform flow as the obstruction height increases from zero. In contrast, the ‘soliton’ wave emerges from a solitary wave solution as the obstruction height grows from zero. These two families of solutions merge at a fold point characterized by a minimum Froude number Fmin . No solutions exist within the Froude number range 1 F  Fmin. Notably, both the 'perturbed' and 'soliton' solutions exhibit wavy-free surfaces.