[Paper Review] Comparison of modulational instabilities in full-dispersion shallow water models

Ashish Kumar Pandey|arXiv (Cornell University)|Aug 1, 2017

Advanced Mathematical Physics Problems10 references3 citations

TL;DR

This paper compares modulational instabilities in full-dispersion shallow water models, including the Whitham, full-dispersion Camassa-Holm (FDCH), and two bi-directional models (FDSW-I and FDSW-II), focusing on how surface tension affects Benjamin-Feir-type instability. It finds that only the FDSW-I model correctly predicts a finite critical wave number in the large surface tension limit, matching the physical water wave problem, while other models fail or diverge.

ABSTRACT

We study the modulational instability of a shallow water model, with and without surface tension, which generalizes the Whitham equation to include bi-directional propagation. Without surface tension, the small amplitude periodic traveling waves are modulationally unstable if their wave number is greater than a critical wave number predicting a Benjamin-Feir type instability and the result qualitatively agrees with the shallow water model in [HP16]. With surface tension, the result qualitatively agrees with the physical problem except for the large surface tension limit which is accurately predicted by the shallow water model in [HP16]. We also compare the results with the Whitham and full-dispersion Camassa-Holm equations.

Motivation & Objective

• To analyze the modulational instability of small-amplitude periodic traveling waves in full-dispersion shallow water models with and without surface tension.
• To compare the stability behavior of different full-dispersion models—Whitham, FDCH, FDSW-I, and FDSW-II—against the physical water wave problem.
• To determine which model most accurately captures the critical wave number for Benjamin-Feir instability across all surface tension regimes, especially in the large surface tension limit.
• To investigate the role of dispersion branches (positive and negative) in modulational instability, particularly in bi-directional models.
• To resolve discrepancies between approximate models and the physical problem in the large surface tension regime, where stability transitions are non-trivial.

Proposed method

• Derives the modulational instability index for each model using a multiple-scales expansion around small-amplitude periodic waves.
• Applies Fourier multiplier operators with symbols derived from the full water wave dispersion relation, including surface tension effects via $ c_{\text{ww}}^2(\kappa) = (1 + T\kappa^2)\frac{\tanh(\kappa)}{\kappa} $.
• Identifies four distinct resonance mechanisms (R1–R4) contributing to instability, with R3 and R4 involving interactions between positive and negative dispersion branches.
• Compares the critical wave number $ \kappa_c(T) $ at which instability occurs across models and the physical problem as a function of surface tension $ T $.
• Uses numerical and asymptotic analysis to evaluate the limit $ \lim_{T \to \infty} \kappa_c(T) $, comparing it to the physical value of approximately 1.121.
• Constructs stability diagrams (Figure 1) showing stable (S) and unstable (U) regions for each model and the physical problem, with labeled curves corresponding to instability mechanisms.

Experimental results

Research questions

• RQ1 How does surface tension affect the critical wave number for modulational instability in full-dispersion shallow water models?
• RQ2 Which of the full-dispersion models—Whitham, FDCH, FDSW-I, or FDSW-II—most accurately reproduces the stability behavior of the physical water wave problem, especially in the large surface tension limit?
• RQ3 Why do some models fail to predict the correct finite limit of the critical wave number as surface tension tends to infinity, while others do?
• RQ4 What role do interactions between positive and negative dispersion branches play in modulational instability, and how do they differ between unidirectional and bi-directional models?
• RQ5 Can the FDSW-I model be considered superior to FDSW-II in capturing physical instability features due to its correct large-$ T $ behavior?

Key findings

• The FDSW-I model correctly predicts that $ \lim_{T \to \infty} \kappa_c(T) \approx 1.054 $, which is finite and close to the physical value of approximately 1.121.
• The Whitham equation fails in the large surface tension limit, as its critical wave number diverges, contradicting the physical problem.
• The FDCH equation improves on the Whitham model, with $ \lim_{T \to \infty} \kappa_c(T) \approx 1.283 $, but still does not match the physical limit.
• For $ 0 < T < 1/3 $, all models agree qualitatively with the physical problem, showing three stable and three unstable intervals of wave number.
• The FDSW-II model incorrectly predicts that all small-amplitude periodic waves are modulationally stable in the large surface tension limit, which is unphysical.
• The FDSW-I model is the only one among the four studied that correctly captures the physical transition from stability to instability at a finite critical wave number in the large surface tension regime.