The evolution of internal wave undular bores: Comparisons of a fully nonlinear numerical model with weakly nonlinear theory

The validity of shallow-water, weakly nonlinear theory for describing the evolution of a single large internal wave depression into an undular bore is explored by comparing theoretical results with results obtained from a fully nonlinear numerical model. inclusion of second-order nonlinear and dispersive terms significantly improves the agreement. Solutions of the KdV and extended KdV equations, which includes second-order nonlinearity, overpredict the wave amplitudes in the undular bore. Inclusion of all second-order nonlinear and dispersive terms significantly improves the predicted amplitudes: however, the resulting evolution equation breaks down for sufficiently large waves. This can be corrected by modifying the linear terms in the equation to give a modified equation. Solutions of this modified second-order equation are in much better agreement with the model results than are the solutions of the KdV equation and the extended KdV equations.