Strong solutions for the Alber equation and stability of unidirectional wave spectra

TL;DR

This paper establishes the first well-posedness theory for the Alber equation, demonstrating linear Landau damping and analyzing stability conditions of ocean wave spectra, with implications for rogue wave formation.

Contribution

It provides the first rigorous mathematical framework for the Alber equation, including well-posedness, stability criteria, and physical implications for ocean engineering.

Findings

Most sea states are stable according to the North Atlantic data

A small fraction (~0.1%) of sea states are modulationally unstable

Unstable states could be precursors to rogue waves

Abstract

The Alber equation is a moment equation for the nonlinear Schr\"odinger equation, formally used in ocean engineering to investigate the stability of stationary and homogeneous sea states in terms of their power spectra. In this work we present the first well-posedness theory for the Alber equation with the help of an appropriate equivalent reformulation. Moreover, we show linear Landau damping in the sense that, under a stability condition on the homogeneous background, any inhomogeneities disperse and decay in time. The proof exploits novel $L^2$ space-time estimates to control the inhomogeneity and our result applies to any regular initial data (without a mean-zero restriction). Finally, the sufficient condition for stability is resolved, and the physical implications for ocean waves are discussed. Using a standard reference dataset (the "North Atlantic Scatter Diagram") it is found that the vast majority of sea states are stable, but modulationally unstable sea states do appear, with likelihood $O(1/1000);$ these would be the prime breeding ground for rogue waves.