Evolution of internal cnoidal waves with local defects in a two-layer fluid with rotation

TL;DR

This paper develops a rotation-augmented weakly nonlinear framework for internal waves in a two-layer fluid by extending the Miyata--Maltseva--Choi--Camassa model (MMCC-f) with Helfrich's f-plane terms. By simultaneously evolving mean fields and deviations with a two-time-scale expansion, the authors bypass the conventional zero-mean constraint and derive a broad class of uni-directional Ostrovsky waves on evolving mean states. They then use pseudospectral Ostrovsky simulations to study how rotation interacts with cnoidal-wave–like initial conditions and local defects, uncovering robust burst phenomena linked to breathers and defect dynamics; expansion defects in particular satisfy all KdV conservation laws exactly. The findings show that rotation can convert localized defects into strong, left-propagating bursts, while pure rotation alone may fail to trigger bursts for some defect types, with implications for energy focusing and rogue-wave formation in geophysical flows. Overall, the work provides a versatile analytical-numerical framework for understanding rotating internal waves near KdV-cnoidal states and highlights the role of defects in burst-generation mechanisms.

Abstract

Internal waves in a two-layer fluid with rotation are considered within the framework of Helfrich's f-plane extension of the Miyata-Maltseva-Choi-Camassa (MMCC) model. Within the scope of this model, we develop an asymptotic procedure which allows us to obtain a description of a large class of uni-directional waves leading to the Ostrovsky equation and allowing for the presence of shear inertial oscillations and barotropic transport. Importantly, unlike the conventional derivations leading to the Ostrovsky equation, the constructed solutions do not impose the zero-mean constraint on the initial conditions for any variable in the problem formulation. Using the constructed solutions, we model the evolution of quasi-periodic initial conditions close to the cnoidal wave solutions of the Korteweg-de Vries (KdV) equation but having a local amplitude and/or periodicity defect, and show that such initial conditions can lead to the emergence of bursts of large internal waves and shear currents. As a by-product of our study, we show that cnoidal waves with expansion defects discussed in this work are generalised travelling waves of the KdV equation: they satisfy all conservation laws of the KdV equation (appropriately understood), as well as the Weirstrass-Erdmann conditions for broken extremals of the associated variational problem and a natural weak formulation. Being smoothed in numerical simulations, they behave, in the absence of rotation, as long-lived states with no visible evolution, while rotation changes this behaviour and leads to the emergence of strong bursts.

Figures (17)

• Figure 1: The first $1.7$ hours of the colour contour time series of temperature profiles off Northern Oregon from the surface to 35m depth. The figure is adapted from SO1998.
• Figure 2: Schematic of a two-layer fluid with rotation in the rigid-lid approximation.
• Figure 3: The effect of rotation on the KdV soliton initial condition in simulations with periodic boundary conditions. First row: view from above (left) and view from below (right) of the interfacial displacement. Second row: interfacial displacement in simulations without the sponge layers (left) and with the sponge layers (right). Third row: comparison of the lead wavepacket in simulations without the sponge layers (black) and with the sponge layers (red dashed).
• Figure 4: The effect of rotation on the KdV cnoidal wave initial condition: view from above (left) and view from below (right) of the interfacial displacement.
• Figure 5: Evolution of the maximum, minimum and amplitude of the interfacial displacement for a soliton (left) and cnoidal wave (right) initial condition on a periodic domain under the effect of rotation.