Short-scale instability of internal waves in rotating stratified fluids
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kakoutas-kakoutas-phd-meche-2026-thesis.pdf
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Author(s)
Kakoutas, Christos
Advisor(s)
Akylas, Triantaphyllos R.
Date Issued
February 2026
Publisher
Massachusetts Institute of Technology
Abstract
Density-stratified fluids, such as the oceans and the atmosphere, support internal gravity waves due to the restoring force of buoyancy. A primary source of internal waves in oceans is the action of the barotropic tide over bottom topography. The wave field generated by this process comprises propagating internal wave beams that interact with the ocean thermocline – the region of most intense stratification, which acts as a waveguide. As a result, part of the tide-induced internal wave energy is deposited in the thermocline in the form of internal wave modes, referred to as internal tides. How internal wave beams and tides disintegrate into smaller scales and eventually get dissipated is of great geophysical interest because this breakdown enhances diapycnal mixing, which affects the global ocean circulation and climate dynamics. A widely studied mechanism of internal wave breakdown is parametric subharmonic instability (PSI). This instability is due to a resonant triad interaction that involves two subharmonic perturbations with half the tide/beam frequency and very fine wavelength, so PSI provides a possible pathway for energy to cascade from large to small scales.
This thesis proposes a short-scale instability mechanism of internal wave beams and tides distinct from PSI. In contrast to earlier studies, which assume resonant triads from the outset, the present work is based on a systematic stability analysis using Floquet theory. The Floquet stability problem is formulated in a Lagrangian frame riding with the basic wave. This eliminates ‘secular’ terms due to the advection of perturbations by the underlying wave and opens the way for solving asymptotically the Floquet problem in the short-scale limit. Overall, the asymptotic analysis reveals that, rather than PSI, small-scale instability is controlled by a novel, multi-scale resonance mechanism which enables transfer of energy to a broader frequency spectrum than just two subharmonics.
Two specific flow configurations of geophysical relevance are studied in detail. (i) The stability to short-scale perturbations of a propagating internal wave mode in a stratified fluid layer bounded by rigid walls -- a simple model of internal tides in the ocean thermocline – including the effects of background rotation and shear flow. It is found that the Lagrangian mean flow associated with the wave mode (tide) is a key factor in deciding whether multi-scale resonance enhances or diminishes the PSI instability growth rates. (ii) Small-scale instabilities of infinitely long, propagating internal wave beams in an unbounded rotating stratified fluid – a model of wave beams in oceans and in the atmosphere. Here, multi-scale resonance instigates the onset of instability at small beam amplitude. Furthermore, formulating the stability problem in the Lagrangian frame establishes the formal connection between the Floquet analysis and the commonly used LG-WKBJ (local) analysis of short-scale wave instabilities.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
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