Interactions between surface gravity wave groups and deep stratification in the ocean - eScholarship

Interactions Between Surface Gravity Wave Groups and Deep Stratification in the Ocean Sean Haney and William Young Scripps Institution of Oceanography, University of California San Diego shaney@ucsd.edu Abstract Groups of surface gravity waves induce horizontally varying Stokes drift which drive con- vergence of water ahead of the group and divergence behind, thereby driving water down- ward in front of the group, and upward in the rear. This “Stokes pumping” creates a deep Eulerian return flow. We assess the impact of stable density stratification on the deep return flow. Our approach is to find solutions of the wave-averaged Boussinesq equations in two (2D) and three dimensions (3D) forced by Stokes pumping at the surface. We find that the shape of the return flow may be changed by vertical density stratification, and if the stratification is sufficiently strong, internal gravity waves may be radiated from the passing surface wave group. In the 2D case, the problem can be solved for arbitrary stratification profiles, however, we find that realistic stratification is too weak to produce internal waves. In the 3D case with constant stratification, internal waves are always emitted by the passing surface wave group. Introduction As surface gravity waves propagate, they induce flow in the direction of propagation at the crest of the waves, and backwards in the troughs. Following a neutrally buoyant particle, or parcel of water, one finds circular orbits, that do not completely close, but rather drift forward. The motion of these particles averaged over a wave period is called the Stokes drift. Mathematically, the Stokes drift is formed by the surface displacement (ζ) dotted into the gradient of the surface wave velocity (u), and averaged over the surface wave period. u S = ζ · ∇u where the overbar denotes an average over the wave phase. The leading order solutions ζ 1 and u 1 = ∇φ are well know (e.g. Phillips, 1977). ζ 1 = 12 a(˜ x, 0) exp ik (x − ct) + c.c. φ = − 2 1 ic a(˜ x, z) exp ik (x − ct) + kz + c.c. , If we insert the well known linear wave solution for ζ and u we find that the Stokes drift is second order in wave slope (ak = ) u S = (ak) 2 ce 2kz where a is the wave amplitude, k is the horizontal wavenumber, and c is the wave phase speed. Now if we consider surface wave groups, the Stokes drift will vary in space and time, that is a = a(x, t). Throughout this work we will use a gaussian envelope as an example, VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016