On the opposing roles of the Boussinesq and non-Boussinesq baroclinic torques in surface gravity wave propagation

Here we suggest an alternative understanding of the surface gravity wave propagation mechanism based on the baroclinic torque, which operates to translate the interfacial vorticity anomalies at the air-water interface. We demonstrate how the non-Boussinesq term of the baroclinic torque acts against the Boussinesq one to hinder wave propagation. By standard vorticity inversion and mirror imaging, we then show how the existence of the bottom boundary affects the two types of torque. Since the opposing non-Boussinesq torque results solely from the mirror image, it vanishes in the deep water limit and its magnitude is half of the Boussinesq torque in the shallow water limit. This reveals that Boussinesq approximation is valid in the deep water limit, even though the density contrast between air and water is large. The mechanistic roles, played by the Boussinesq and non-Boussinesq parts of the baroclinic torque, remain obscured in the standard derivation where the time-dependent Bernoulli equation is implemented instead of the interfacial vorticity equation. Finally, we note on passing that the Virial theorem for surface gravity waves can be obtained solely from considerations of the dynamics at the air-water interface.