Modelling of Polarity Change in a Nonlinear Internal Wave Train in Laoshan Bay

There are now several observations of internal solitary waves passing through a critical point where the coefficient of the quadratic nonlinear term in the variable coefficient Korteweg-de Vries equation changes sign, typically from negative to positive as the wave propagates shoreward. This causes a solitary wave of depression to transform into a train of solitary waves of elevation riding on a negative pedestal. However, recently a polarity change of a different kind was observed in Laoshan Bay, China, where a periodic wave train of elevation waves converted to a periodic wave train of depression waves as the thermocline rose on a rising tide. This paper describes the application of a newly developed theory for this phenomenon. The theory is based on the variable coefficient Korteweg-de Vries equation for the case when the coefficient of the quadratic nonlinear term undergoes a change of sign and predicts that a periodic wave train will pass through this critical point as a linear wave, where a phase change occurs that induces a change in the polarity of the wave, as observed. A two-layer model of the density stratification and background current shear is developed to make the theoretical predictions specific and quantitative. Some numerical simulations of the variable coefficient Korteweg-de Vries equation, and also the extended variable coefficient Korteweg-de Vries equation, are reported that confirm the theoretical predictions and are in good agreement with the observations.