An Unstable Mode Formed by Combination of Kelvin-Helmholtz Waves and Tollmien-Schlichiting Waves in Jets on the Bottom

Instability of jets on the bottom, such as those formed under large-amplitude lee waves or overflows (density currents), is one of mixing processes in bottom boundary layers. Numerical experiments by Abe and Nakamura (2013) showed the two types of shear instabilities in jets on the bottom, that is, Kelvin-Helmholtz (KH) waves and Tollmien-Schlichiting (TS) waves. Their wavelengths and phase speeds were almost the same, suggesting that KH and TS waves combine to form another unstable mode, which we call combined mode hereafter. However, no studies have dealt with the interaction between these instabilities.

In this study, we consider the possible mechanism of the combination of KH and TS waves and confirm the combination mechanism by numerical experiments using a two-dimensional model.

Development of KH waves is explained by interaction of a pair of vorticity waves propagating in opposite directions on two vorticity interfaces. Similarly, TS waves consist of two modes interacting with each other: a vorticity wave propagating on a vorticity interface and a viscous mode near the bottom, which arises to satisfy a no-slip condition at the bottom. Combining these, we propose the mechanism of a new unstable mode in jets on the bottom, which have two vorticity interfaces in the shear layer; that is, if all of the two waves on the interfaces and the viscous mode on the bottom have the same phase speed, these interact with each other, and the combined mode may develop. In the experiment where the background shear flow had an idealized form of that in the experiments of Abe and Nakamura (2013), the two vorticity waves and the viscous mode grew simultaneously with their phase relations that agreed with the proposed mechanism. Moreover, the wavelength and phase speed of the fastest growing perturbations were consistent with the KH and TS waves in the experiments of Abe and Nakamura (2013). These facts suggest that combined modes could affect mixing near the bottom.