Interference of internal waves due to two point vortices: linear analytical solution and nonlinear interaction

In this work, we consider steady two-dimensional interfacial waves in a two-layer stratified fluid, which is induced by a vortex pair located in the lower layer of the fluids. A mathematical model based on the boundary integral equation method and the potential-flow theory is established. The linear analytical solution for the linearized model is given in the form of Cauchy integral and then asymptotic behaviour for large x is presented. The fully nonlinear model is solved by the Jacobian-free Newton-Krylov (JFNK) method numerically. Nonlinear characteristics of wave profiles are identified compared with the linear results under different vortex strengths and the distance between the vortex pair. The amplitude of steady downstream waves is found to vary periodically with respect to the distance of the vortex pair, which can be regarded as the interference between waves produced by each vortex. For equal-strength counter- and co-rotating pairs, the downstream wave heights of linear solutions can be eliminated for some special values of the distance between point vortices, namely, the destructive interference occurs. Meanwhile, the wave only exists between the vortex pair like trapped waves. So does the nonlinear counterpart for counter-rotating pairs, but it could not be diminished with any distance.