Cherenkov interaction of vortices with a free surface

The interaction of vortex filaments in an ideal incompressible fluid with t he free surface of the latter is investigated in the canonical formalism. A Hamiltonian formulation of the equations of motion is given in terms of bo th canonical and noncanonical Poisson brackets. The relationship between th ese two approaches is analyzed. The Lagrangian of the system and the Poisso n brackets are obtained in terms of vortex lines, making it possible to stu dy the dynamics of thin vortex filaments with allowance for finite thicknes s of the filaments. For two-dimensional flows exact equations of motion des cribing the interaction of point vortices and surface waves are derived by transformation to conformal variables. Asymptotic steady-state solutions ar e found for a vortex moving at a velocity lower than the minimum phase velo city of surface waves. It is found that discrete coupled states of surface waves above a vortex are possible by virtue of the inhomogeneous Doppler ef fect. At velocities higher than the minimum phase velocity the buoyant rise of a vortex as a result of Cherenkov radiation is described in the semicla ssical limit. The instability of a vortex filament against three-dimensiona l kink perturbations due to interaction with the "image'' vortex is demonst rated. (C) 1999 American Institute of Physics. [S1063-7761(99)00903-8].