NONLINEAR-INTERACTION OF SHEAR-FLOW WITH A FREE-SURFACE

In this paper the nonlinear evolution of two-dimensional shear-flow in stabilities near the ocean surface is studied. The approach is numeric al, through direct simulation of the incompressible Euler equations su bject to the dynamic and kinematic boundary conditions at the free sur face. The problem is formulated using boundary-fitted coordinates, and for the numerical simulation a spectral spatial discretization method is used involving Fourier modes in the streamwise direction and Cheby shev polynomials along the depth. An explicit integration is performed in time using a splitting scheme. The initial state of the flow is as sumed to be a known parallel shear flow with a flat free surface. A pe rturbation having the form of the fastest growing linear instability m ode of the shear flow is then introduced, and its subsequent evolution is followed numerically. According to linear theory, a shear flow wit h a free surface has two linear instability modes, corresponding to di fferent branches of the dispersion relation: Branch I, at low wavenumb ers; and Branch II, at high wavenumbers for low Froude numbers, and lo w wavenumbers for high Froude numbers. Our simulations show that the t wo branches have a distinctly different nonlinear evolution. Branch I: At low Froude numbers, Branch I instability waves develop strong oval -shaped vortices immediately below the ocean surface. The induced velo city field presents a very sharp shear near the crest of the free-surf ace elevation in the horizontal direction. As a result, the free-surfa ce wave acquires steep slopes, while its amplitude remains very small, and eventually the computer code crashes suggesting that the wave wil l break. Branch II: At low Froude numbers, Branch II instability waves develop weak vortices with dimensions considerably smaller than their distance from the ocean surface. The induced velocity field at the oc ean surface varies smoothly in space, and the free-surface elevation t akes the form of a propagating wave. At high Froude numbers, however, the growing rates of the Branch II instability waves increase, resulti ng in the formation of strong vortices. The free surface reaches a lar ge amplitude, and strong vertical velocity shear develops at the free surface. The computer code eventually crashes suggesting that the wave will break. This behaviour of the ocean surface persists even in the infinite-Froude-number limit. It is concluded that the free-surface ma nifestation of shear-flow instabilities acquires the form of a propaga ting water wave only if the induced velocity field at the ocean surfac e varies smoothly along the direction of propagation.