The stability of surface currents to small perturbations is Investigated, with the purpose to assess the effect of the free surface on the form of the instability waves. The equations of motion and continuity are thus linearized around the time-average flow, assumed to have a self-similar form. The linearized kinematic and dynamic boundary conditions are satisfied on the ocean surface. For wavy, perturbations, requiring that the linearized problem has a non-trivial solution defines the dispersion relation of the flow, which relates the frequency and wavenumber of the wave. The dispersion relation is solved numerically to give the frequency as a function of the wavenumber. A range of unstable waves is found, in the sense that real wavenumbers yield complex frequencies with positive imaginary parts. Examination of the double roots of the dispersion relation reveals however that the Instability IS of the convective type. Consequently spatially growing waves are possible, which are determined numerically, using an Iterative procedure. It is shown that, even though the presence of the fr.ee surface has at low Froude numbers very little influence on the dispersion relation, compared to the one obtained in unbounded fluid, it drastically alters the form of the instability wave by destroying the axisymmetry of the flow, and imposing instead a plane symmetry around the centerplane of the current. As a result the instability modes are not helical, as in unbounded fluids. The Visualization of this patterns on the ocean surface in particular is reminiscent of the instability wave observed in a two-dimensional flow having its shear in the horizontal plane, even though the assumed basic state is equally sheared along the vertical and the horizontal plane.
Ocean currents develop large-scale meandering motions which lead to the spin-off of vortices from the current.
