## Transcritical flow of a stratified fluid: The forced extended Korteweg-de Vries model

Transcritical, or resonant, flow of a stratified fluid over an obstacle is studied
using a forced extended Korteweg - de Vries model. This model is particularly relevant
for a two-layer fluid when the layer depths are near critical, but can also be useful in
other similar circumstances. Both quadratic and cubic nonlinearities are present and they
are balanced by third order dispersion. We consider both possible signs for the cubic
nonlinear term but emphasise the less-studied case when the cubic nonlinear term and the
dispersion term have the same-signed coefficients. In this case, our numerical
simulations show that two kinds of solitary waves are found in certain parameters
regimes. One kind is similar to those of the well-known forced Korteweg - de Vries
model and occurs when the cubic nonlinear term is rather small, while the other kind is
irregularly generated waves of variable amplitude, which may continually interact. To
explain this phenomenon, we develop a hydraulic theory in which the dispersion term in
the model is omitted. This theory can predict the occurrence of upstream and
downstream undular bores, and these predictions are found to agree quite well with the
numerical simulations.

### History

#### School

- Science

#### Department

- Mathematical Sciences