On the nonlinear critical layer below a nonlinear unsteady surface wave

Abstract

A theory is presented for the development of the nonlinear critical layer below an unsteady free surface wave, of amplitude e, described by the Korteweg-de Vries (KdV) equation. The problem is formulated (via the Euler equations) for wave propagation over an arbitrary shear in a two-dimensional channel which contains a critical level. The equations are scaled so as to be valid in the far-field regime of the surface wave, appropriate to the existence of the KdV equation, i.e. long waves. The regions above and below the critical layer are solved (to O(e), as e → 0) and thence expanded in the neighbourhood of the critical layer itself. The symmetry of the critical layer solution, assuming that it exists, is then sufficient to determine the Burns integral for the linearized wave speed, and the relevant KdV equation. These turn out to be the classical results evaluated in terms of finite parts.The critical layer, of thickness O(e½), is analysed to O(e) and matched to the outer regions of the flow field. The initial configuration is taken to contain no closed streamlines, and so the vorticity can, presumably, be assigned from the undisturbed conditions at infinity. The initial surface profile must therefore contain a single peak, but by virtue of the KdV equation this can evolve into any number of solitons. Between consecutive pairs of peaks there will now appear regions of closed streamlines (cat's-eyes) with known vorticity. No recourse to a viscous argument is necessary to uniquely determine this vorticity. However, it is shown that the vorticity cannot be prescribed arbitrarily at all orders, initially: the long-wave assumption imposes a certain structure on the problem, and then the continuity of stream function and particle velocity fixes the vorticity. This agrees with the work of Varley & Blyth (1983) on the hydraulic equations. The vorticity inside the separating streamlines is obtained to O(e), but it is shown that for unsteady motion this asymptotic expansion is not uniformly valid as the bounding streamlines are approached. An alternative method, which exploits Varley & Blythe's approach, is used to confirm the correctness of our results away from these boundaries, and to indicate that a non-uniformity is present near the separating streamlines. Thus the model requires the inclusion of a vortex sheet; for steady flow a jump in vorticity is sufficient. The removal of the discontinuity by allowing a distortion of the main flow outside the critical layer is briefly discussed.Some results are presented for the formation of a single cat's-eye by using the exact 2-soliton solution of the KdV equation.