Deep Flows Transmitted by Forced Surface Gravity Waves

Abstract

We examine a two-dimensional deep-water surface gravity wave packet generated by a pressure disturbance in the Lagrangian reference frame. The pressure disturbance has the form of a narrow-banded weakly nonlinear deep-water wave packet. During forcing, the vorticity equation implies that the momentum resides entirely in the near-surface Lagrangian-mean flow, which in this context is often called the “Stokes drift”. After the forcing turns off, the wave packet propagates away from the forcing region, carrying with it most of the energy imparted by the forcing. These waves together with their induced long wave response have no momentum in a depth integrated sense, in agreement with the classical results of Longuet-Higgins and Stewart (Deep Sea Research and Oceanographic Abstracts 11, 592−562) and McIntyre (Journal of Fluid Mechanics 106, 331−347). The total flow associated with the propagating packet has no net momentum. In contrast with the finite-depth scenario discussed by McIntyre (Journal of Fluid Mechanics 106, 331−347), however, momentum imparted to the fluid during forcing resides in a dipolar structure that persists in the forcing region—rather than being carried away by shallow-water waves. We conclude by examining waves propagating from deep to shallow water and show that wave packets, which initially have no momentum, may have non-zero momentum in finite-depth water through reflected and trapped long waves. This explains how deep water waves acquire momentum as they approach shore. The artificial form of the parameterized forcing from the wind facilitates the thought experiments considered in this paper, as opposed to striving to model more realistic wind forcing scenarios.