The nonlinear evolution of internal tides. Part 2. Lagrangian transport by periodic and modulated waves

Abstract

We examine Lagrangian transport by a nonlinearly evolving vertical mode-1 internal tide in non-uniform stratification. In a companion paper (Sutherland & Dhaliwal, J. Fluid Mech., 2022, in press) it was shown that a parent internal tide can excite successive superharmonics that superimpose to form a solitary wave train. Despite this transformation, here we show that the collective forcing by the parent wave and superharmonics is effectively steady in time. Thus we derive relatively simple formulae for the Stokes drift and induced Eulerian flow associated with the waves under the assumption that the parent waves and superharmonics are long compared with the fluid depth. In all cases, the Stokes drift exhibits a mixed mode-1 and mode-2 vertical structure with the flow being in the waveward direction at the surface. If the background rotation is non-negligible, the vertical structure of the induced Eulerian flow is equal and opposite to that of the Stokes drift. This flow periodically increases and decreases at the inertial frequency with maximum magnitude twice that of the Stokes drift. When superimposed with the Stokes drift, the Lagrangian flow at the surface periodically changes from positive to negative over one inertial period. If the background rotation is zero, the induced Eulerian flow evolves non-negligibly in time and space for horizontally modulated waves: the depth below the surface of the positive Lagrangian flow becomes shallower ahead of the peak of the amplitude envelope and becomes deeper in the lee of the peak. These predictions are well-captured by fully nonlinear numerical simulations.

The nonlinear evolution of internal tides. Part 1: the superharmonic cascade

“…However, Baker & Sutherland (2020) showed that ultimately, such a steady state does not evolve from a horizontally periodic internal mode: starting with no superharmonics, the parent wave excites internal waves that grow and decay periodically in amplitude, provided that the amplitude of the parent wave is sufficiently small. In our companion paper (Sutherland & Yassin 2022), we likewise show that ultimately, a steady state does not evolve from horizontally modulated internal modes. This occurs because the natural frequency, , of the superharmonic wave is nearly double the frequency, , of the parent wave.…”

“…Future theoretical work will relax the assumption of constant and using the Wentzel–Kramers–Brillouin (WKB) approximation, and the influence of the finite spanwise extent of internal tide beams will also be considered. In a companion paper, we relax the assumption of periodicity of the internal tide to examine a spatially modulated internal tide both in theory and simulations (Sutherland & Yassin 2022). Therein it is shown that the self-interaction of the internal tide and its superharmonics leads to a near-steady forcing of an induced Eulerian flow whose magnitude remains sufficiently small to have negligible influence upon the evolution of the internal tide and its superharmonics.…”

“…Unlike the theory, the simulations exhibited small-scale oscillations about the predicted parent wave amplitude. These were due predominantly to weak interactions between the parent wave and the induced Eulerian flow, which has a mixed mode-1/mode-2 structure (Sutherland & Yassin 2022). The results show that for the moderately small value of in this simulation, only the - and -superharmonics grow to significant amplitude, and the amplitude of the parent wave decreases only slightly by the time the superharmonics have grown to their largest amplitude at .…”

“…In this work, we ignore the self-interaction of waves leading to an induced Eulerian flow (Bühler 2014;van den Bremer, Yassin & Sutherland 2019). This is because, as is shown in our companion paper (Sutherland & Yassin 2022), superharmonics are near-resonant with the parent wave, whereas the induced Eulerian flow is not. Hence this flow has negligible influence upon the parent wave and its superharmonics.…”

The effect of model tidal forcing on virtual particle dispersion and accumulation at the ocean surface

“…Tidal currents have also been considered using the SMOC dataset (Drillet et al, 2019) to understand the trajectories of Sargassum in the Tropical Atlantic by van Sebille et al (2021). Moreover, the Lagrangian transport due to internal tides has been studied by Sutherland and Yassin (2022), but they found it to be negligible when averaged over an inertial period. Technological advances in the past years have allowed recent OGCMs to include tidal forcing, improving the representation of internal waves and tide fields (Le Sommer et al, 2018).…”