Surface Gravity Wave Groups Drive "Stokes Pumping" and Radiate Internal Gravity Wave Wakes

Surface gravity wave groups induce time and horizontally varying mass flux. As wave groups pass by the mass flux is larger in the center of the group than in front or behind. This mass flux divergence induces “Stokes pumping”: downward flow ahead of the group and upward flow behind, creating a deep Eulerian return flow. Panel (a) of the figure shows the stream function for the deep return flow in an unstratified ocean beneath a surface wave group with gaussian amplitude modulation (the amplitude of the plotted surface waves is exaggerated for ease of visualization). The return flow extends thousands of meters (many times the wave group width) deep while the Stokes drift and surface wave orbital velocities extend at most a few tens of meters. In the stratified case, the Eulerian return flow wiggles isopycnals beneath the mixed layer, thereby radiating a wake of internal gravity waves. The stratified case is shown in panel (b) of the figure (dashed contours indicate negative values of the stream function), where the same wave group propagates over an ocean with constant stratification (buoyancy frequency, N = 7 × 10⁻³s⁻¹). The radiated internal waves have frequencies near, but below, the buoyancy frequency, and wavenumbers near, but below, the maximum wavenumber given by 2N/c, where c/2 is the group speed of the surface waves.

We solve the wave-averaged Boussinesq equations in two and three dimensions (2D and 3D respectively), and find that these two cases are distinct. In 2D internal waves will not radiate unless the stratification is very strong, or the surface waves are very slow. This is because the internal wave phase is set by the Stokes pumping. Therefore, the internal wave phase speed must match the surface wave group speed. In 2D, this can only be achieved with relatively fast internal waves (strong stratification), or slow surface waves. However, in 3D, internal waves are always radiated because the internal waves may propagate obliquely to the surface wave propagation direction, allowing their phase speed in that direction to be much faster than the phase speed perpendicular to the internal wave crests. This constraint allows us to define a maximum wake angle such that sin(θ_max) = 2Nh/nπc, where h is the depth of the ocean.

To assess the impact of this phenomenon we compute the energy flux from surface to internal waves. We find that, in 3D, the energy flux scales with the fourth power of the buoyancy frequency and surface wave amplitude, and inversely with the fifth power of the surface wave period. For characteristic values of surface wave parameters (e.g. 8 second period and wave heights of a few meters) and buoyancy frequency (2000 second buoyancy period), the energy flux from surface to internal waves is small compared to the total amount of energy in a surface wave group. However, Stokes pumping may be a significant source of energy for internal waves with a spectral peak near the buoyancy frequency.