2. Preliminaries

To describe the two-dimensional water flows under investigation we choose rectangular coordinates $(x,y)$ with the horizontal $x$ -axis in the direction of wave propagation and the $y$ -axis vertical. The motion is assumed periodic in the $x$ -direction, with wavelength $L$ , but will not be periodic in the time variable $t$ . The origin is taken to be on the mean surface level. Since we are interested in large-scale surface deep-water waves, it is reasonable, cf. Longuet-Higgins & Cokelet (Reference Longuet-Higgins and Cokelet1976) and Cokelet (Reference Cokelet1977), to consider the fluid inviscid, homogeneous and incompressible. The fluid motion is initiated from rest by conservative forces, so that it will be irrotational at all times prior to breaking. In particular, the slow diffusion of vorticity inwards from the boundaries is neglected. This theoretical set-up for the study of breaking was proposed by Longuet-Higgins & Cokelet (Reference Longuet-Higgins and Cokelet1976), being pursued in Peregrine, Cokelet & McIver (Reference Peregrine, Cokelet, McIver and Edge1980) and in many subsequent numerical investigations. Experimental confirmation of its practical relevance is provided in Melville, Veron & White (Reference Melville, Veron and White2002).

Let us now present the governing equations. The equation of motion in the bulk of the fluid is Euler’s equation

(2.1a ) $$\begin{eqnarray}\displaystyle u_{t}+uu_{x}+vu_{y}=-\frac{1}{{\it\rho}}P_{x},\end{eqnarray}$$

(2.1b ) $$\begin{eqnarray}\displaystyle v_{t}+uv_{x}+vv_{y}=-\frac{1}{{\it\rho}}P_{y}-g,\end{eqnarray}$$

for the velocity field $(u,v)$ , where $g$ is the gravitational constant of acceleration, ${\it\rho}$ is the (constant) density of water and $P$ is the pressure. In addition, the equation of mass conservation

(2.2) $$\begin{eqnarray}u_{x}+v_{y}=0\end{eqnarray}$$

and the constraint of irrotational flow

(2.3) $$\begin{eqnarray}u_{y}=v_{x}\end{eqnarray}$$

Figure 1.

Depiction of a typical deformation of a deep-water wave profile at successive times as it steepens and overturns, taking for initial conditions a symmetric travelling wave. The relative size of the overhanging jet distinguishes before breaking between the two breaker types, with small and moderate corresponding to a spilling and plunging breaker, respectively. Experimental data and numerical simulations indicate the location $\boldsymbol{\times }$ of the point where the maximal horizontal fluid velocity at the surface is attained.

should hold within the fluid. Since the dynamics of deep-water waves is unaffected by the bottom topography, for simplicity we consider a flat bed $y=-d$ . This assumption is not merely a mathematical convenience since abyssal plains are found in all major sea and ocean basins, covering more than 30 % of the Earth’s surface (about the same as all the exposed land combined). On the flat impermeable bed we impose the kinematic boundary condition

(2.4) $$\begin{eqnarray}v=0\quad \text{on }y=-d.\end{eqnarray}$$

To capture overhanging wave profiles, we represent at a fixed time $t$ the free surface $S(t)$ parametrically as

(2.5a,b ) $$\begin{eqnarray}x={\it\alpha}(s,t),\quad y={\it\beta}(s,t),\end{eqnarray}$$

where ${\it\alpha}$ and ${\it\beta}$ are smooth functions with the property that, for every fixed $t$ , the map $s\mapsto ({\it\alpha}(s,t)-s,{\it\beta}(s,t))$ is periodic with wavelength $L$ and such that ${\it\alpha}_{s}^{2}(s,t)+{\it\beta}_{s}^{2}(s,t)>0$ for every $s\in [0,L]$ ; the last condition ensures regularity by preventing the existence of singular points. For example, overhanging waves with a profile having per wave period one ascending part and a declining face (common for waves close to breaking, as depicted in figure 1) correspond to ${\it\beta}_{s}(s,t)>0$ for $s\in (s_{0}(t),s_{+}(t))$ and ${\it\beta}_{s}(s,t)<0$ for $s\in (s_{+}(t),s_{0}(t)+L)$ , where $s_{0}(t)$ and $s_{+}(t)$ identify the location of the wave trough and wave crest, respectively. On the free surface the dynamic boundary condition

(2.6) $$\begin{eqnarray}P({\it\alpha}(s,t),{\it\beta}(s,t),t)=P_{atm},\end{eqnarray}$$

where $P_{atm}$ is the constant atmospheric pressure, decouples the motion of the water from that of the air above it. We can ignore the airflow above the water since the density of water is $10^{3}$ times that of air, while surface tension effects are neglected in order to concentrate on the large-scale features of the water waves. Finally, the condition that the free surface is an interface is expressed by requiring that the flow of fluid relative to the free boundary must be tangential to it. The normal to the free surface in space–time being given by the vector product of the tangential vectors $({\it\alpha}_{s},{\it\beta}_{s},0)$ and $({\it\alpha}_{t},{\it\beta}_{t},1)$ , we can express the vanishing of the component of fluid velocity normal to the free boundary as the kinematic boundary condition

(2.7) $$\begin{eqnarray}u{\it\beta}_{s}-v{\it\alpha}_{s}+{\it\alpha}_{s}{\it\beta}_{t}-{\it\beta}_{s}{\it\alpha}_{t}=0\quad \text{on the free surface}~S(t).\end{eqnarray}$$

Equations (2.1a )–(2.7) represent the governing equations that we will work with. The existence of smooth solutions as long as the free surface is non-self-intersecting is granted; see the discussion in Constantin (Reference Constantin2011). Our aim is to investigate the time evolution of the maximum of the horizontal fluid velocity. Note that we cannot rely on the usual approximate theories for the dynamics of surface waves (the small-amplitude linear theory, the nonlinear shallow-water theory, or variants thereof) since they cease to be valid for overhanging wave profiles.

3. Main result

We are interested in the time evolution of the maximum of the horizontal fluid velocity. At any fixed time $t$ , this maximum $M(t)$ is attained only along the free surface. Indeed, at a fixed time $t$ , $(x,y)\mapsto u(x,y,t)$ is a harmonic function in the fluid domain ${\it\Omega}(t)$ , due to (2.2) and (2.3). Since $u$ is not constant, its maximum can only be attained on the boundary. Were $(x_{0},-d)$ such a location on the flat bed, at time $t_{0}$ , this would entail $u_{y}(x_{0},-d,t_{0})<0$ by Hopf’s maximum principle, see Constantin (Reference Constantin2011). But then (2.3) forces $v_{x}(x_{0},-d,t_{0})<0$ , in contradiction to (2.4). Thus

(3.1) $$\begin{eqnarray}M(t)=\sup _{(x,y)\in {\it\Omega}(t)}\{u(x,y,t)\}=\sup _{s\in [0,L]}\{u({\it\alpha}(s,t),{\it\beta}(s,t),t)\},\quad t\in [0,T),\end{eqnarray}$$

where $T>0$ is the breaking time. We claim that

the function  $t\mapsto M(t)$  is absolutely continuous and therefore almost everywhere differentiable, with

(3.2) $$\begin{eqnarray}M^{\prime }(t)=-\frac{1}{{\it\rho}}P_{x}(X(t),t)\quad \mathit{for~almost~every}~t\in (0,T),\end{eqnarray}$$

where  $X(t)=(x(t),y(t))$  is any location where  $u$  attains its maximum.

To prove the validity of (3.2), we first show that $M$ is absolutely continuous with

(3.3) $$\begin{eqnarray}M^{\prime }(t)=u_{t}(X(t),t)+u_{x}(X(t),t){\it\alpha}_{t}({\it\xi}(t),t)+u_{y}(X(t),t){\it\beta}_{t}({\it\xi}(t),t)\end{eqnarray}$$

for almost every $t\in (0,T)$ , where ${\it\xi}(t)$ is such that $X(t)=({\it\alpha}({\it\xi}(t),t),{\it\beta}({\it\xi}(t),t))$ . Subsequently, we use the fact that

(3.4) $$\begin{eqnarray}u_{x}(X(t),t){\it\alpha}_{s}({\it\xi}(t),t)+u_{y}(X(t),t){\it\beta}_{s}({\it\xi}(t),t)=0,\end{eqnarray}$$

since $s={\it\xi}(t)$ is a maximum of the function $s\mapsto u({\it\alpha}(s,t),{\it\beta}(s,t),t)$ . In combination with (2.7) and (2.1a ), we check that this will lead us from (3.3) to (3.2). Before providing the details, let us comment on the likelihood of (3.3), drawing attention to some delicate related issues. The existence of ${\it\xi}(t)$ being granted, if the function $t\mapsto {\it\xi}(t)$ were differentiable, then the function $t\mapsto u({\it\alpha}({\it\xi}(t),t),{\it\beta}({\it\xi}(t),t),t)$ could be differentiated and (3.3) would emerge by the chain rule, if we take (3.4) into account. However, this approach cannot be validated since $t\mapsto {\it\xi}(t)$ might be multi-valued or, even if it is single-valued, might fail to be continuous. To see this, perform localized uncorrelated time-dependent perturbations near the two peaks at $s_{-}=0$ and $s_{+}=L$ of the periodic function $s\mapsto \cos (2{\rm\pi}s/L)$ in the periodicity window $[0,L]$ to set up a scenario in which the location of the maximum in one wave cycle is always at ${\it\xi}(t)=s_{+}$ for $t>0$ , another in which the location is at all times $t>0$ both at ${\it\xi}(t)=s_{-}$ and ${\it\xi}(t)=s_{+}$ (and therefore undetermined), and also a scenario in which the location alternates regularly between the two points $s_{\pm }$ , jumping from one to the other after it becomes undetermined. Nevertheless, the evaluation of $u({\it\alpha}(\cdot ,t),{\it\beta}(\cdot ,t),t)$ at any point ${\it\xi}(t)$ where the maximum is attained produces an absolutely continuous function $M(t)$ that satisfies (3.3), a situation reminiscent of that investigated by Constantin & Escher (Reference Constantin and Escher1998) in a more abstract setting. To see why this is so, note first that $M(t)$ is locally Lipschitz on $[0,T)$ . Indeed, given ${\it\varepsilon}>0$ , if $t,t^{\prime }\in [0,T-{\it\varepsilon}]$ are such that, say, $M(t)\geqslant M(t^{\prime })$ , then

(3.5) $$\begin{eqnarray}\displaystyle 0 & {\leqslant} & \displaystyle M(t)-M(t^{\prime })=u({\it\alpha}({\it\xi}(t),t),{\it\beta}({\it\xi}(t),t),t)-u({\it\alpha}({\it\xi}(t^{\prime }),t^{\prime }),{\it\beta}({\it\xi}(t^{\prime }),t^{\prime }),t^{\prime })\nonumber\\ \displaystyle & {\leqslant} & \displaystyle u({\it\alpha}({\it\xi}(t),t),{\it\beta}({\it\xi}(t),t),t)-u({\it\alpha}({\it\xi}(t),t^{\prime }),{\it\beta}({\it\xi}(t),t^{\prime }),t^{\prime })\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \sup _{x\in [0,L]}\{|u({\it\alpha}(x,t),{\it\beta}(x,t),t)-u({\it\alpha}(x,t^{\prime }),{\it\beta}(x,t^{\prime }),t^{\prime })|\}\leqslant k_{{\it\varepsilon}}|t-t^{\prime }|\end{eqnarray}$$

by the mean-value theorem, with

(3.6) $$\begin{eqnarray}\displaystyle k_{{\it\varepsilon}}=\sup _{{\it\chi}\in [0,L],{\it\tau}\in [0,T-{\it\varepsilon}]}\{|u_{x}(\mathfrak{X}({\it\chi},{\it\tau}),{\it\tau}){\it\alpha}_{t}({\it\chi},{\it\tau})+u_{y}(\mathfrak{X}({\it\chi},{\it\tau}),{\it\tau}){\it\beta}_{t}({\it\chi},{\it\tau})+u_{t}(\mathfrak{X}({\it\chi},{\it\tau}),{\it\tau})|\}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where we have denoted $\mathfrak{X}({\it\chi},{\it\tau})=({\it\alpha}({\it\chi},{\it\tau}),{\it\beta}({\it\chi},{\it\tau}))$ . Locally Lipschitz functions being absolutely continuous, they are differentiable almost everywhere, cf. the discussion in Constantin (Reference Constantin2011). Since we cannot rely on the chain rule to compute $M^{\prime }(t)$ , we argue as follows. Given $t\in (0,T)$ and $h\in (0,T-t)$ , we have that $(M(t+h)-M(t))/h$ equals

(3.7) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{u({\it\alpha}({\it\xi}(t+h),t+h),{\it\beta}({\it\xi}(t+h),t+h),t+h)-u({\it\alpha}({\it\xi}(t),t),{\it\beta}({\it\xi}(t),t),t)}{h}\nonumber\\ \displaystyle & & \displaystyle \quad \geqslant \frac{u({\it\alpha}({\it\xi}(t),t+h),{\it\beta}({\it\xi}(t),t+h),t+h)-u({\it\alpha}({\it\xi}(t),t),{\it\beta}({\it\xi}(t),t),t)}{h}.\end{eqnarray}$$

In the limit $h\downarrow 0$ the right-hand side of the above estimate converges to the limit

(3.8) $$\begin{eqnarray}\displaystyle \mathfrak{L}(t)=u_{x}(X(t),t){\it\alpha}_{t}({\it\xi}(t),t)+u_{y}(X(t),t){\it\beta}_{t}({\it\xi}(t),t)+u_{t}(X(t),t). & & \displaystyle\end{eqnarray}$$

Consequently $\liminf _{h\downarrow 0}((M(t+h)-M(t))/h)\geqslant \mathfrak{L}(t)$ . On the other hand, $(M(t)-M(t-h))/h$ equals

(3.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{u({\it\alpha}({\it\xi}(t),t),{\it\beta}({\it\xi}(t),t),t)-u({\it\alpha}({\it\xi}(t-h),t-h),{\it\beta}({\it\xi}(t-h),t-h),t-h)}{h}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \frac{u({\it\alpha}({\it\xi}(t),t),{\it\beta}({\it\xi}(t),t),t)-u({\it\alpha}({\it\xi}(t),t-h),{\it\beta}({\it\xi}(t),t-h),t-h)}{h}.\end{eqnarray}$$

In the limit $h\downarrow 0$ the right-hand side of the above estimate converges to $\mathfrak{L}(t)$ , so that $\limsup _{h\downarrow 0}((M(t)-M(t-h))/h)\leqslant \mathfrak{L}(t)$ . At an instant $t\in (0,T)$ when $M$ happens to be differentiable we have $M^{\prime }(t)=\liminf _{h\downarrow 0}((M(t+h)-M(t))/h)=\limsup _{h\downarrow 0}((M(t)-M(t-h))/h)$ . This proves the validity of (3.3). To complete the checking of (3.2), let us assume that $M$ is differentiable at the instant $t$ and that ${\it\alpha}_{s}({\it\xi}(t),t)\neq 0$ . Dividing (3.4) by ${\it\alpha}_{s}({\it\xi}(t),t)$ and multiplying the outcome by ${\it\alpha}_{t}({\it\xi}(t),t)$ , by taking (2.7) into account, we get

(3.10) $$\begin{eqnarray}\displaystyle & & \displaystyle u_{x}(X(t),t){\it\alpha}_{t}({\it\xi}(t),t)+u_{y}(X(t),t){\it\beta}_{t}({\it\xi}(t),t)\nonumber\\ \displaystyle & & \displaystyle \quad =u_{y}(X(t),t)v(X(t),t)-u(X(t),t)u_{y}(X(t),t)\frac{{\it\beta}_{s}({\it\xi}(t),t)}{{\it\alpha}_{s}({\it\xi}(t),t)}.\end{eqnarray}$$

Using now (3.4), equations (2.1a ) and (3.3) lead us to the desired relation (3.2).