NONLINEAR EFFECTS IN WAVE SCATTERING AND GENERATION Flow Interaction with Topography

When a uid ow interacts with a topographic feature, and the uid can support wave propagation, then there is the potential for waves to be generated upstream and/or downstream. In many cases when the topographic feature has a small amplitude the situation can be successfully described using a linearised theory, and any nonlinear e ects are determined as a small perturbation on the linear theory. However, when the ow is critical, that is, the system supports a long wave whose group velocity is zero in the reference frame of the topographic feature, then typically the linear theory fails and it is necessary to develop an intrinsically nonlinear theory. It is now known that in many cases such a transcritical, weakly nonlinear and weakly dispersive theory leads to a forced Kortewegde Vries (fKdV) equation. In canonical form, this is given by ut ux + 6uux + uxxx + fx = 0, where u(x; t) is the amplitude of the critical mode, t is the time coordinate and x is the spatial coordinate, is the phase speed of the critical mode, and f(x) is a representation of the topographic feature. In this article we shall sketch the contexts where the fKdV equation is applicable, and describe some of the most relevant solutions. There are two main classes of solutions. In the rst, the initial condition for the fKdV equation is u(x; 0) = 0 so that the waves are generated directly by the ow interaction with the topography. In this case the solutions are characterised by the generation of upstream solitary waves and an oscillatory downstream wavetrain, with the detailed structure being determined by and the polarity of the topographic forcing term f(x). In the second class a solitary wave is incident on the topography, and depending on the system parameters may be repelled with a signi cant amplitude change, trapped with a change in amplitude, or allowed to pass by the topography with only a small change in amplitude.