Benjamin-Feir Instability of Interfacial Gravity-Capillary Waves in a Two-Layer Fluid. Part I

TL;DR

This paper analyzes the modulational stability of interfacial gravity-capillary waves in a two-layer fluid, revealing complex stability structures influenced by layer properties, and connects these findings to known semi-infinite layer configurations.

Contribution

It provides a comprehensive stability diagram for two-layer interfacial waves considering finite thicknesses and surface tension, highlighting new stability structures and physical mechanisms.

Findings

Identification of characteristic stability structures such as localized loops and corridors.

Influence of layer asymmetry on the stability regions and their connectivity.

Connection of finite-layer results with semi-infinite layer configurations.

Abstract

This study presents a detailed investigation of the modulational stability of interfacial wave packets in a two-layer inviscid incompressible fluid with finite layer thicknesses and interfacial surface tension. The stability analysis is carried out for a broad range of density ratios and geometric configurations, enabling the construction of stability diagrams in the (\rho,k) -plane, where \rho is the density ratio and k is the carrier wavenumber. The Benjamin-Feir index is used as the stability criterion, and its interplay with the curvature of the dispersion relation is examined to determine the onset of modulational instability. The topology of the stability diagrams reveals several characteristic structures: a localized loop of stability within an instability zone, a global upper stability domain, an elongated corridor bounded by resonance and dispersion curves, and a degenerate cut structure arising in strongly asymmetric configurations. Each of these structures is associated with a distinct physical mechanism involving the balance between focusing/defocusing nonlinearity and anomalous/normal dispersion. Systematic variation of layer thicknesses allows us to track the formation, deformation, and disappearance of these regions, as well as their merging or segmentation due to resonance effects. Limiting cases of semi-infinite layers are analyzed to connect the results with known configurations, including the `half-space--layer', `layer--half-space', and `half-space--half-space' systems. The influence of symmetry and asymmetry in layer geometry is examined in detail, showing how it governs the arrangement and connectivity of stable and unstable regions in parameter space.