Dynamics of ideal fluid flows over an uneven bottom

Abstract

Two problems of the stability of ideal fluid flows over an uneven bottom are considered. The first is the study of stratified flow with a ‘rigid lid’. We use the method of multiple scales to derive an equation describing the evolution of internal waves corresponding to different modes and wave vectors. For the case of sinusoidal bottom irregularities we have constructed a solution describing the increase in time of the internal wave field - this proves the instability of the basic flow. The phenomenon is interpreted as a result of interaction (mutual generation) of internal waves with energies of opposite signs. Our consideration is based on the Hamiltonian approach which enables us to prove in the most simple way the existence of waves carrying negative energy. The case of random (not sinusoidal) bottom irregularities is also studied. Using the kinetic equation for the amplitudes of internal waves derived in the paper, we have established that the basic flow remains unstable as well. In the second part of the paper we consider the homogeneous flows with a free upper boundary. It is shown that this problem can be reduced to the previous one, with the only difference being that the role of unstable perturbations is now played by the surface (not internal) gravity waves. The Hamiltonian approach is consistently applied and allows us to take into account the nonlinearity of waves.