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

Area of Science:

• Fluid dynamics
• Nonlinear wave phenomena
• Stratified fluid mechanics

Background:

• Interfacial waves in stratified fluids are crucial in oceanography and geophysics.
• Vortex dynamics significantly influence fluid interfaces.
• Understanding wave generation mechanisms is key to predicting environmental phenomena.

Purpose of the Study:

• To investigate steady two-dimensional interfacial waves in a two-layer stratified fluid.
• To model waves induced by a vortex pair in the lower fluid layer.
• To analyze both linear and fully nonlinear wave behaviors.

Main Methods:

• Development of a mathematical model using boundary integral equation method and potential-flow theory.
• Derivation of a linear analytical solution involving Cauchy integrals.
• Numerical solution of the fully nonlinear model using the Jacobian-free Newton-Krylov (JFNK) method.

Main Results:

• Wave profiles exhibit nonlinear characteristics compared to linear solutions.
• Downstream wave amplitude shows periodic variation with vortex pair distance, indicating wave interference.
• Destructive interference can eliminate downstream waves in linear models for specific vortex arrangements, creating trapped waves.

Conclusions:

• The study provides insights into wave generation and propagation in stratified fluids driven by vortices.
• Nonlinear effects can alter wave behavior, particularly for counter-rotating vortex pairs.
• Periodic amplitude variations highlight the complex interplay between vortex configurations and wave dynamics.