Ocean models often simplify continuous density stratification into two discrete layers: a warm layer on top, a cold layer below, with a sharp interface between them. The bilayer model is computationally tractable and captures the dominant mode of internal wave propagation. The assumption: as the actual stratification becomes sharper — the pycnocline thins — the bilayer model becomes more accurate.
The paper tests this assumption by taking the sharp-stratification limit of the full Euler equations and comparing to both the bilayer Euler equations and the bilayer shallow-water equations. For the linearized system without shear, convergence holds — the bilayer model correctly captures the limiting behavior.
With shear flows, convergence fails. Kelvin-Helmholtz instabilities — interfacial perturbations amplified by the velocity difference across the interface — grow without bound as the stratification sharpens. The bilayer model predicts specific wave propagation; the full model predicts instability. The instabilities are not artifacts of the continuous formulation — they're real physics that the bilayer model suppresses by replacing a thin but finite transition zone with a mathematical discontinuity.
The finite thickness of the pycnocline regularizes the Kelvin-Helmholtz instability. The bilayer limit removes this regularization. The limit is singular: approaching it doesn't produce convergence but instability, because the regularizing mechanism — diffusive smoothing across the interface — vanishes exactly when it's needed most. The bilayer model is accurate precisely in the regime where it's unnecessary (no shear) and breaks down precisely where it would be useful (with shear).