Internal Tides: Instabilities and Lagrangian Transport

In uniform stratification, a horizontally propagating, vertically bounded internal tide eventually gives up energy to subharmonics through triad resonant instability (TRI), which is a generalization of parametric subharmonic instability. In non-uniform stratification, however, TRI is suppressed. But in this case the internal tide immediately excites superharmonics with double the horizontal wavenumber of the internal tide and nearly double the frequency. Particularly in the tropics, where the Coriolis parameter is small, the superharmonics are nearly resonant with the internal tide, growing to large amplitude and themselves exciting superharmonics. This work will present theory, in the form of coupled ordinary differential equations, which predict that the superharmonic cascade leads to the formation of a solitary wave-train. The results are in excellent agreement with fully nonlinear numerical simulations. For long waves in strong near-surface stratification, the results agree well with the prediction of shallow water theory including rotation through the Ostrovsky equation. The theory also predicts Eulerian transport by the internal tide which, together with the Stokes drift, predicts the Lagrangian transport. On the f-plane, the Lagrangian flow oscillates at the Coriolis frequency with zero net transport. Near the equator, the Eulerian flow grows over time exhibiting complex vertical structure.