Fold catastrophe in breaking waves

TL;DR

The paper develops a slow-fast dynamical-systems framework for gravity-driven wave breaking by tracking an energetic surface hotspot in 2D potential flow. It reveals a fold catastrophe in the $(m,A)$ plane that governs approach to breaking, with fast evolution of the surface slope $m$ and slow variation of the focusing parameter $A$, leading to finite-time blowup when the fold is crossed. A threshold slope $ heta^*= ext{arctan}( ext{√2}-1)=22.5^ ext{o}$ demarcates breaking inception via maximal anisotropy between velocity and pressure Hessians, and a gravity-analogue black hole geometry (apparent and event horizons) provides a unifying interpretive framework. The results connect to prior energetic-hotspot and inflection-point studies, quantify crest-height limits through kinetic-over-potential energy surpluses, and suggest extensions to 3D, viscosity, and surface tension for improved rogue-wave predictions. Overall, the fold-catastrophe perspective offers a geometric, first-principles account of wave steepening and breaking with potential implications for ocean-wave modeling and hazard assessment.

Abstract

We present a dynamical-systems perspective on wave breaking for ideal incompressible free-surface flows. By tracking the most energetic hotspot on the wave surface, we find that near breaking the surface slope m evolves on a fast timescale governed by the small parameter epsilon = (partial_z u)^(-1), the inverse vertical velocity gradient at the hotspot, while the focusing parameter A = (U - Ce)/(U - Creq) varies slowly and adiabatically. Here U is the horizontal fluid velocity at the energetic point, Ce its propagation speed, and Creq the equivalent crest speed. This slow-fast structure reveals a fold catastrophe in the (m, A) space whose boundary forms the geometric skeleton organizing the dynamics near breaking. Finite-time blowup occurs when the trajectory crosses this boundary, marking the onset of breaking. The inception of breaking is further characterized by crossing the slope threshold theta* = arctan(sqrt(2) - 1) = 22.5 degrees. This critical angle marks the maximum anisotropy that can be sustained between the Hessians of the velocity and pressure fields, reflecting an imbalance between kinetic and potential energy fluxes. The anisotropy of the velocity Hessian also gives rise to the classical 30-degree slope observed at the inflection point of steep waves near breaking inception. The crest height is limited by the maximum excess of kinetic over potential energy that the flow can sustain, beyond which breaking becomes inevitable. Wave breaking can also be interpreted as a gravity analogue of a collapsing black hole, with apparent and event horizons representing the onset and inception of breaking.

Figures (8)

• Figure 1: Top panel: snapshots of the (black) surface elevation $\eta(x,t)$ and (red) kinetic energy $\tilde{k}_e(x,t)$ of the largest non-breaking wave (labeled $2$) in the simulated right-going wave group $C3N5A0.514$BarthelemyJFM2018 at the pre-focusing ($P$), focusing ($F$), and post-focusing ($R$) instants, in the middle top panel of Fig. \ref{['FIG3fold']}. The energetic hotspot (red square) is located at the front of the wave before focusing and moves to the back after passing through the crest (blue square). The inflection point is also depicted (blue diamond). Bottom panel: (left) time profiles of the horizontal velocity gradient $U_z$ tracked at the energetic hotspot of the four major successive waves of the simulated right-going group $C3N5A0.514$BarthelemyJFM2018. These are labeled as $0,1,2,3$ with varying thickness to denote the increasing amplitude, and wave $3$ is breaking; (right) relational fast time $\tau$ as function of the absolute time $t$.
• Figure 2: Top panels: time profiles of the wave parameters $m$, $s$ and $R=m+s$ tracked at the energetic hotspot of the four major successive waves of the simulated right-going group $C3N5A0.514$BarthelemyJFM2018. Bottom panels: same for the slow parameters $A$ and $\beta=C_e/U-1$ tracked at the energetic hotspot. Waves are labeled as $0,1,2,3$ with varying thickness to denote the increasing amplitude, and with wave $3$ breaking. Characteristic slopes are also shown (red lines) and quantified in terms of $\epsilon=1/U_z\approx 0.1$.
• Figure 3: Fold catastrophe for inviscid and irrotational waves in deep water. (Left) fold boundary $\partial\mathcal{F}_0$ in the $(m,A)$ plane and trajectories (solid lines) of the four major successive waves of the simulated right-going wave group $C3N5A0.514$BarthelemyJFM2018. These are labeled as $0,1,2,3$ at the beginning (early time) of the trajectories, and depicted with increasing thickness. The thickest (dark red) line indicates the largest breaking wave. Points $P$, $F$, and $R$ mark the pre-focusing, focusing, and post-focusing phases of the largest non-breaking wave ($2$) (see Fig.\ref{['FIG2waveprofiles+crestgroupspeeds']}). Theoretical predictions (dashed lines) from the slow-fast model are also depicted. The path of a linear packet is also reported (amplitude $\mu = k a = 0.1$ and spectral bandwidth $\nu = 0.3$). (Left) the desymmetrized half-domain $m\le0$.
• Figure 4: Wave dynamics in the $(s,m)$ domain. Trajectories (black lines) of the four major successive waves in the simulated right-going wave group $C3N5A0514$, reported in BarthelemyJFM2018, are shown. These trajectories are labeled $0,1,2,3$ at their initial points (before focus) and are depicted with increasing thickness. The thickest (red) line corresponds to the breaking wave. The curves $\varGamma_1=\{(s,m):\tan\theta^+_{U} + m=0 \rightarrow s=4(-m+m^3)/(1-6m^2+m^4)\}$ and $\varGamma_2 =\{(s,m):m_2=0\rightarrow s=-m\}$, delimit the wave dynamics between the asymptotic slopes $\theta_*=\pm22.5^{\circ}=\pm\arctan(\sqrt{2}-1)$. The curve $\varGamma_*={(s,m):s=(-m+5m^3)/(1-6m^2+m^4)}$ represents the approximate maximally non-breaking (marginal) wave. The breaking wave (red) exceeds the $\theta_*$ threshold, marking the onset of breaking inception.
• Figure 5: Eigen-structure of the weight matrices $G_u$ and $G_p$ associated with the Hessians of the fluid velocity $U$ and pressure $P$, respectively. Top panels: (Left) slope angle $\theta^U_+$ of the eigenvector of the velocity weight matrix $G_u$ corresponding to the positive eigenvalue $\lambda^U_+$ as a function of the surface slope $m$, for the four major successive waves of the simulated right-going wave group $C3N5A0514$BarthelemyJFM2018. (Right) Same, but for the slope angle $\theta^P_+$ of the eigenvector of the pressure weight matrix $G_P$ associated with the positive eigenvalue $\lambda^P_+$. The breaking wave is shown as the thickest dark-red line. Bottom panels: (Left) Ratio $\lambda_+^U/\lambda_+^P$ of the positive eigenvalues, and (right) ratio $\lambda_-^U/(-\lambda_-^P)$ of the negative eigenvalues. The waves are labeled $0,1,2,3$ at the beginning (before focus) of their trajectories and are depicted with increasing thickness. The breaking wave is the thickest curve, colored distinctly from the non-breaking waves. The non-breaking waves are confined within the shaded region bounded by two limiting cases: (i) the condition $\tan\theta^U_{+}+m=0$, and (ii) vanishing momentum parameter $m_2=0$ (curves $\varGamma_1$, $\varGamma_2$ in Fig. \ref{['ms_domain']}). The dotted line represents the approximately maximally non-breaking (marginal) wave (curve $\varGamma^*$ in Fig. \ref{['ms_domain']}). The breaking wave surpasses the region bounded by the surface angle $\theta_*=\pm22.5^{\circ}=\pm\arctan(\sqrt{2}-1)$ (dashed lines), marking the inception of breaking.