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# On some properties of the pressure in nonlinear two-dimensional potential flow with a free surface

Yves-Marie Scolan

► To cite this version:

Yves-Marie Scolan. On some properties of the pressure in nonlinear two-dimensional potential flow with a free surface. Applied Ocean Research, 2024, 150, pp.104123. 10.1016/j.apor.2024.104123. hal-05340848

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https://hal.univ-brest.fr/hal-05340848v1

Submitted on 31 Oct 2025

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Distributed under a Creative Commons Attribution 4.0 International License
On some properties of the pressure in nonlinear
twodimensional potential flow with a free surface.

Yves-Marie SCOLAN

ENSTA Bretagne, IRDL UMR 6027,
2 Rue François Verny 29806 Brest Cedex 9¹

**Abstract**

Two-dimensional nonlinear free surface flows are nowadays routinely simulated on com-
puters. In the context of the potential flow theory, fine details of the free surface motion
can be captured. However, few studies focus on the flow beneath the free surface. The
novelty of the present work lies in identifying lines where the pressure gradient is colinear
with one eigenvector of the pressure Hessian matrix. The spatial and temporal evolution
of these lines offers new insights into the complexity of highly nonlinear flows. This pres-
sure characteristic is used to revisit four more or less recent research studies on breaking
waves.

Keywords: Potential flow theory, Nonlinear free surface flow, Twodimensional flow, Breaking wave, Hessian of pressure.

**1. Introduction**

The pressure analysis in Stokes waves has been the subject of much research. Theoretical results are detailed in Constantin and Strauss (2010) or Clamond (2012), among others. Experimental data are shown in Umeyama (2017). However, both theoretical and experimental results concern weakly nonlinear waves for which the pressure mainly reaches its maximum at the bottom. But the pressure field can have large variations inside the fluid and close to the free surface. The so-called Taylor's condition (see Taylor, 1950) allows for the exhibition of highly accelerated flow if a maximum of pressure occurs in the vicinity of the free surface. The resulting phenomenon can sometimes be interpreted as a singularity in finite time (see Hogrefe *et al*, 1998; Pomeau *et al*, 2008). This is the case of the so-called flip-through discovered by Cooker and Peregrine (1990) and commented on by Liu and Pego (2019). Some works also focus on the curvature singularities on the free surface profile. For example Baker and Xie (2011) revisit the idealized model that describes the tip of the crest in a plunging breaker (see Longuet-Higgins, 1980a).

One of the aims of the present study is to propose a new technique for locating singularities on the free surface. This technique is based on the analysis of the pressure and its derivatives. To the best of the author's knowledge, the literature does not provide such a detailed analysis of pressure prior to the appearance of a singularity.

<footnote>Email address: yves-marie.scolan@ensta-bretagne.fr (Yves-Marie SCOLAN)
ENSTA Bretagne, IRDL UMR 6027,
2 Rue François Verny 29806 Brest Cedex 9)</footnote>
1.1. Physical configuration and method of solution

The fluid is contained in a rectangular tank with width L; at rest, the fluid height is h. The flow is described in a two-dimensional space with the Cartesian coordinates (x, y) and time t. The origin of the coordinate system is centered at the corner left bottom as illustrated in Fig. 1. The complex coordinate z = x + iy (with i² = -1)defines

<figure><img src="image_3.png" /><figcaption>Figure 1: Notations for the description of the free surface flow in a rectangular tank with width L filled with a fluid up to the height h.</figcaption></figure>

a point in this physical plane. The gravity acts along the negative y direction. We numerically simulate two-dimensional nonlinear free surface flows in potential theory. Laplace's equation is solved using the so-called desingularized technique proposed by Tuck (1998). The velocity potential is represented by a set of N Rankine sources. These singularities are placed outside the fluid domain so that an integral equation is avoided. To account for the solid and impermeable walls of the tank, the conformal mapping w = f_m(z) = - cos πz / L turns the semi-infinite vertical strip with width L into the half space ℑ(w) > 0. As a result, by using the mirror image of the singularities with respect to the real axis in the transformed plane, the impermeability condition is easily accounted for. The fluid flow is described in the coordinate system (0, x, y). This is a non-Galilean coordinate system if the tank is moving. The complex potential F that describes the fluid flow in the coordinate system (0, x, y) reads

$$
F(z,t) = \sum_{j=1}^{N} q_j(t)G(w, W_j(t)), \quad \text{with} \quad w = f_m(z), \tag{1}
$$

where the intensity of the source # j is q_j and W_j = f_m(Z_j) is the complex coordinate of
the source # j in the transformed plane w. The function G denotes the complex potential
of the combined influence of two sources located at complex coordinates W_j and W&#773;_j (the
overline denotes the complex conjugate). This Green function G satisfies a homogeneous
Neumann boundary condition all over the walls of the tank. The characteristics (q_j, Z_j)
of the N sources are computed so that the free surface is an isobaric and material line.
The surface tension effects are discarded. If the tank is moving, the flow described in
a Galilean coordinate system, would require the computation of the so-called Stokes-
Joukowski potentials (see Faltinsen et al., 2000). Some free surface flows described below
are generated by a forced horizontal motion of the reservoir. If the horizontal velocity is
denoted u_h(t), the total complex potential will contain F and the additional contribution
u_h(t)z.
The velocity potential is transported by N Lagrangian markers that are located at the actual free surface. The two free surface boundary conditions are integrated over time with a fourth-order Runge-Kutta scheme. Neither regridding, nor smoothing is performed during the numerical simulations. At each time step of the simulation, the position of the free surface markers and the associated velocity potential are recorded. In addition, the characteristics of the Rankine sources (position $Z_j$ and intensity $q_j$ and their time derivative) are also recorded. This is sufficient to compute the pressure and all its derivatives a posteriori.

Theorem of convergence of the desingularized technique do not exist for nonlinear free surface flows. Convergence criteria are established in Cao et al (1991); those are used in the present numerical applications. For linear configurations (fluid domain independent on time) the convergence is theoretically studied in Scolan (2015). The main numerical parameters are the distance of desingularization, the number of singularities and the time step. In all the simulations presented here, these parameters are chosen in such a way as to ensure satisfactory conservation of mass and energy. In the sequel, the errors are mentioned for each studied numerical case. More importantly, a special attention is paid to computing accurately the partial time derivative of the potential. Indeed, this is necessary to make sure that the spatial derivatives of the pressure are computed with enough accuracy up to the free surface; this numerical aspect of the problem is not detailed in the present paper.

Other numerical techniques exist to solve the same problem and simulate the same nonlinear free surface flows studied in this paper. The purpose of the present paper is not to contextualize the desingularized technique in the numerical techniques used in potential theory. A benchmark (see Benoit et al, 2018) shows that the desingularized technique provides the same results than other potential flow solvers, at least for the propagation of a wave train over a very long distance. Comparisons between a Navier-Stokes solver and the present potential flow solver (see Etienne et al, 2018) show an accurate computation of the energies for a more complicated configuration consisting of a two fluid system.

## 1.2. Governing equations for the pressure

The pressure $p(x, y, t)$ in the fluid follows from Bernoulli's equation written below in its complex form

$$ \frac{p}{\rho} = -\Re \left( F_{,t} + \frac{1}{2} |F_{,z}|^2 + (\dot{u}_h - ig)z \right) + c(t), \quad (2) $$

where $c(t)$ is a function of time only. The fluid density is $\rho$ and $g$ denotes the acceleration of gravity. The dot denotes the time derivative, $|\cdot|$ is the complex conjugate and the real part of a complex number is denoted $\Re$. The pressure gradient $\nabla p = (p_{,x}, p_{,y})$ is directly obtained from Euler's equation also written below in its complex form

$$ F_{,zt} + F_{,zz} \overline{F_{,z}} + \frac{p_{,z}}{\rho} - ig + \dot{u}_h = 0, \quad p_{,z} = p_{,x} - ip_{,y} \quad (3) $$

In equations (2) and (3), the term $\dot{u}_h$ is the acceleration of the known forced horizontal motion of the reservoir.

The elements of the Hessian matrix of the pressure are

$$ \mathbf{H} = \begin{pmatrix} p_{,xx} & p_{,xy} \\ p_{,xy} & p_{,yy} \end{pmatrix}, \quad (4) $$
<figure><img src="image_4.png" /><figcaption>Figure 2: Geometrical illustration of the backbone line defined on the pressure surface $p(x, y)$. Blue lines: elemental parabolic arc-lengths with negative curvature in a plane containing the eigenvector $\vec{e}_2$. Red line: backbone line where $\vec{\nabla}p$ is colinear to $\vec{e}_1$. Point H: pressure surface approximated as an hyperbolic paraboloid in a plane containing the pressure gradient. Point E: pressure surface approximated as an elliptic paraboloid in a plane containing the pressure gradient.</figcaption></figure>

they are high derivatives (up to third order) of the potential.

$$ \left\{ \begin{array}{l} p_{,xx} = -f - \Re(A) \\ p_{,yy} = -f + \Re(A) \\ p_{,xy} = \Im(A) \end{array} \right. , \quad \text{with} \quad \left\{ \begin{array}{l} f = \rho |F_{,zz}|^2 \\ A = \rho (F_{,zzz} F_{,z} + F_{,zzt}) \end{array} \right. \qquad (5) $$

Note that the desingularized technique makes the computation of **H** straightforward with the same precision as for the velocity potential. The eigenvalues $\lambda$ and the eigenvectors $\vec{v}$ of the Hessian matrix **H**($x, y, t$) are denoted

$$ \lambda = -f \pm |A|, \quad \vec{e} = \begin{pmatrix} \Im(A) \\ \Re(A) \pm |A| \end{pmatrix}. \qquad (6) $$

The determinant of **H** is known as the Gaussian curvature of the pressure

$$ \Omega = f^2 - |A|^2 \qquad (7) $$

This quantity is more informative than the mean curvature given by the pressure Laplacian $\Delta P = -2f$; this mean curvature is indeed always negative while $\Omega$ changes sign when $f = |A|$. One eigenvalue (say $\lambda_2$) is always negative. The eigenvector associated to the eigenvalue $\lambda_1$ that can change sign is denoted $\vec{e}_1 = (e_x, e_y)$. In the following, we shall call the line composed of points where the pressure gradient $\vec{\nabla}p$ is colinear with $\vec{e}_1$ the backbone line. This occurs when $e_x p_{,y} = e_y p_{,x}$ alternatively when

$$ \Im(A)p_{,y} - (\Re(A) + |A|)p_{,x} = 0 \qquad (8) $$

Fig. 2 illustrates this line geometrically. The local expansion of the pressure as a Taylor development up to second order leads to determining the principal directions along which the pressure surface can be approximated, either as an elliptic paraboloid (point E) or as an hyperbolic paraboloid (point H) in a tangent plane containing the pressure gradient. In Fig. 2, the elemental elliptic arc-lengths (blue) illustrate the local negative curvature associated to the second eigenvector $\vec{e}_2$; this vector is orthogonal to the first eigenvector $\vec{e}_1$. It is clear that the inner product of the pressure gradient with $\vec{e}_2$ is nil at the intersection of the elemental elliptic arc-lengths and the red line. To the author's knowledge, there are no bibliographical references that introduce the backbone line as used in this paper.
Theoretically, the backbone line can be described implicitly with Cartesian coordinates $(x, y)$ provided that a starting point is known beforehand. This point can be located at the bottom of the tank where a local maximum necessarily exists if the pressure is mainly hydrostatic and the fluid kinematics or dynamics are weak. This maximum is hence beneath a crest. However, several lines can co-exist in the fluid domain, and sometimes one of the lines can end inside the fluid far from the bottom. That is why in practice, we determine this line by computing $\vec{e}_1$ and $\nabla p$ over a structured mesh. The finer the mesh, the finer the determination of the points where $e_x p_{,y} = e_y p_{,x}$. This technique requires identifying precisely the line of interest and then focusing on the region where it appears. The retained points are rearranged and interpolated to describe the line with an arc-length. Some parts of the line of interest are embedded in regions where the pressure gradient vanishes. This does not mean that fluid kinematics and dynamics are low. It should be noted that the backbone line contains the points where the pressure gradient drops to zero; these are either a saddle point or a maximum of the pressure. Anticipating the occurrence of a pressure maximum close to the free surface is a crucial issue; indeed, this is a premonitory signal of a local high acceleration as well.

In the present study, waves are generated either by shaking the reservoir (section §3), by simulating a dam-break flow (section §4), or by initializing with another code solution (section §5). There also exist exact solutions of the problem. The present theoretical findings are tested on these analytical solutions (section §2). Supplementary materials are available. These describe the videos illustrating the application cases developed in sections §3 and §5. For the three numerical simulations described in the paper, the initial conditions (position of the markers and the distribution of velocity potential at the initial free surface) are available on request.

## 2. Analytical exact solution

The pioneering works by Longuet-Higgins (1980a) provide exact solutions of the two boundary conditions for the pressure $p$ and its Lagrangian derivative $dp/dt$ (time differentiation following a fluid particle), which both vanish at the free surface. The obtained solutions are valid locally and describes the fluid kinematics and dynamics in the close vicinity of a wave crest that mimics actual overturning wave crest. One of the exact solutions, known as the hyperbola solution, follows from the choice of the complex potential $F(z, t) = \frac{1}{2}az^2$. It is expressed in terms of the complex coordinates $z = x + iy$ and factor $a$, which is a complex function of time $t$ only. The origin of the coordinate system $z = 0$ is in free fall, and gravity does not appear in the equations (Euler and Bernoulli). The inner angle $\gamma$ of the two asymptotes of the hyperbola is such that $\cos \gamma = |a|^2/|\dot{a}|$, where $\dot{a}$ denotes the time derivative of $a$. The angle between the bisector of the two asymptotes and the horizontal axis $y = 0$ is $\delta$, with $\delta = 1/2 \arg(\dot{a}) \pm \pi/2$. When looking for the pressure gradient in the fluid (in absence of gravity), by using the idealized complex potential above, the Euler equation yields $p_{,x} - ip_y = -(\dot{a}z + |a|^{2\bar{z}})$. After some algebra, the Cartesian coordinates of $\vec{e}_1$ satisfy the equation $e_y/e_x = \frac{\Re(\dot{a})+|\dot{a}|}{\Im(\dot{a})}$. The line corresponding to the colinearity of the pressure gradient and the eigenvector $\vec{e}_1$ of the pressure Hessian matrix is defined by the equation

$$e_x p_{,y} - e_y p_{,x} = -2 \sin \delta |\dot{a}| (|\dot{a}| + |a|^2) \Im(ze^{i\delta}) = 0, \quad (9)$$
<figure><img src="image_5.png" /><figcaption>Figure 3: Hyperbola solution of the nonlinear free surface boundary condition proposed by Longuet-Higgins (1980a). The asymptotes (black lines) makes an inner angle $γ$ such that $\cos \gamma = 3/5$. The inclination is $\delta = \pi/4$. The blue line corresponds to the points where $e_{x}p_{,y} - e_{y}p_{,x} = 0$. The vector field represents the normalized pressure gradient $\frac{\nabla p}{||\nabla p||}$.</figcaption></figure>

This quantity vanishes along the bisector of the two asymptotes; therefore, this line intersects the tip of the crest, as illustrated in Fig. 3. The vector field of the normalized pressure gradient is also plotted. This vector field emanates normally from the free surface and reaches the backbone line. This theoretical result shows that the local curvature of the free surface plays an important role in the existence of the backbone line. If the radius of curvature of the free surface is positive, one can expect that a backbone line to emanate from the point where the radius of curvature is the smallest. This feature is independent of the gravity; thus, it is likely to occur in a plunging breaker.

### 3. Critical jet in a plunging breaker

We revisit the parametric analysis documented in Scolan (2018). A rectangular tank with length $L$ is filled up to the depth $h \approx L/4$. The tank is forced to move horizontally with a cycling motion. The amplitude and period of the motion are chosen so that we always end up with an overturning crest before the end of the cycling motion of the tank. In the present case, the resulting flow also leads to the onset of a small critical jet below the tip of the overturning crest. An example of critical jet is shown in Fig. 4. This discovery (see Scolan *et al*, 2016) was the starting point of the present pressure analysis. Indeed, questions arise: why does this critical jet appear? what are the fluid kinematics and dynamics that precede this appearance? Theoretical elements of an answer are proposed by Longuet-Higgins (1980b); he asserts that tracking the pressure maximum is a key aspect of the problem. In fact, this jet starts to develop as soon as a local pressure maximum occurs inside the fluid near the free surface. The reason is simple: a vanishing pressure gradient close to the free surface, where the pressure drops to zero necessarily, yields a high pressure gradient at the free surface and therefore a high Lagrangian acceleration as well. This is true whatever the value of the maximum of pressure provided that it is not zero, obviously. This is not allowed in the present free surface flow since the Laplacian of
the pressure (known as the mean curvature) is negative throughout the fluid domain. It
remains to track the zeroes of the pressure gradient. When progressing in that direction,
we discover that tracking the zeroes of the quantity $e_x p_{,y} - e_y p_{,x}$ can provide even more
information. This quantity also vanishes with the pressure gradient. The final finding is
the fact that the zeroes of $e_x p_{,y} - e_y p_{,x}$ define continuous lines that emanate from the free
surface and can possibly reach the bottom. In the present configuration, there is one line
that links the tip of the critical jet to a point inside the fluid. This is true largely before
its appearance and up to the onset of numerical instabilities that the numerical model
cannot handle. For this simulation, the errors on mass and energy conservation are at
worst $10^{-4.6}$ and $10^{-3.8}$ respectively.

The parametric study mentioned above consists of using a reference simulation that
leads to a critical jet as shown in Fig. 4. By making restarts at intermediate instants of
this reference simulation, in a now fixed tank, thus fixing the level of energy (potential
and kinetic) injected into the fluid, we can capture more or less prominent jets. Here we
focus on the biggest simulated jet.

Fig. 4a shows the initial free surface profile and the one at which we analyze the flow.
The close-up in Fig. 4a shows the details of the tips (main crest and critical jet). Fig.
4b shows how the normalized pressure gradient normally emanates from the free surface
and joins the central line, mimicking the backbone of a fish. The flow is studied here at
an instant when two disconnected backbone lines coexist. We focus on the backbone line
that joins the tip of the critical jet to a point inside the fluid. This continuous line is
described with an arc-length. Fig. 5 shows the variations of the pressure gradient along
the backbone line starting from the lowest point (marked with a bullet). The dynamics of
the fluid are important in the vicinity of the critical jet, and gravity does not play a role in
this region. The two contributions of the Lagrangian acceleration ($\frac{d\vec{u}}{dt} = \vec{u}_t + (\vec{u} \cdot \nabla)\vec{u}$) can each reach more than 150 times the acceleration of gravity, but these two vectors are
opposite in direction, yielding a vertical Lagrangian acceleration which is still about 20
times the acceleration of gravity. The fluid kinematics are also important; the velocity
at the tip of the critical jet reaches between 2 and 3 times the velocity calculated at the
tip of the main crest. It is worth noting that the vertical component of the pressure
gradient vanishes at the tip of the critical jet, whereas the horizontal component drops
to a small value as well. This means that the vertical Lagrangian acceleration in this jet
reduces to gravity; in other words, the fluid particles are almost in free fall even if the jet
turns in the opposite direction of the gravity. Actually, some numerical experiments in
the modelling of such a jet (see Scolan, 2022) lead to the conclusion that the critical jet
always turns on the side where the Lagrangian acceleration is the highest. Indeed, in the
present simulation, the maximum acceleration occurs along the part of the free surface
bounded by the two tips. This point is marked with a blue circle in Fig. 4b where the
Lagrangian acceleration is about 140 times the acceleration of gravity.

Tracking the backbone line is illustrated in the supplementary material with 4 videos.
(backbone-critical-jet-plunging-breaker-v1.avi, backbone-critical-jet-plunging-breaker-v2.avi,
backbone-critical-jet-plunging-breaker-v3.avi and pressure-critical-jet-plunging-breaker.avi).
This gives more insight into the time evolution of the backbone line that is often embedded
in a region of low pressure gradient. In particular, we can easily track the occurrence of
the pressure maximum where the pressure gradient vanishes. The onset of a local pressure
maximum in the vicinity of the free surface triggers the cusp at the free surface and the
subsequent critical jet.

## 4. Dam-break flow leading to a breaking soliton

The dam-break flow is documented in Scolan and Etienne (2023). This configuration is revisited here to analyze the continuous change between the Stokes' critical value of $\frac{p_y}{\rho g} = -1/2$ (see Longuet-Higgins, 1963) and the free fall dynamic in the overturning wave where the pressure gradient must vanish. Only the upper part of the crest is of interest. The pressure field is hence analyzed early enough before the vertical front appears and up to the development of the plunging breaker. For this simulation, the errors on the mass and energy conservation are at worst $10^{-5}$ and $10^{-4.5}$ respectively. The selected free surface profiles are plotted in Fig. 6.

Fig. 7a shows the initial and the last computed free surface profiles. At this last computed time, the vector field of the pressure gradient is calculated at the crest, and it is plotted in Fig. 7b. Here again, we observe how the pressure gradient emanates normally from the free surface to join the backbone line, which is superimposed. In the present case, this line joins the tip of the overturning crest to the bottom of the tank.

The temporal-spatial variation of the pressure gradient along the backbone line is shown in Fig. 8 a, b, c. The two Cartesian components have roughly the same variation but not in the same intervals. Two stages are identified : before and after the first occurrence of the vertical front along the free surface. In Figs. 6 and 8, the black line corresponds to this vertical front of the crest, which occurs at instant $t = 1.043$s. Below this instant, the variation of $\frac{p_y}{\rho g}$ along the backbone changes significantly over a short interval of time; in the present case $t \in [0.979 : 1.029]$s. In this time interval, the vertical pressure gradient $\frac{p_y}{\rho g}$ passes the threshold $-1/2$. Below this threshold $\frac{p_y}{\rho g}$ roughly decreases towards more negative values as we approach the free surface. About this threshold, there a transition region where $\frac{p_y}{\rho g}$ has two extrema : one minimum and one maximum. These two extrema are identified in Fig. 8 b, c with blue dots. At the last stage, roughly $t \approx 1.029$s, these two extrema coincide at an inflection point. Above this threshold, the variation of $\frac{p_y}{\rho g}$ along the backbone line is monotonous; as expected, it increases towards zero close to the free surface.

The threshold predicted by the Stokes' theory is hence quite reasonable. However, the changes of monotonicity of the pressure gradient occur before the crest's front becomes vertical. This characteristics better anticipates the onset of breaking.

Later (after the onset of the vertical front at the crest) when the plunging breaker is increasing in size, the pressure gradient has another variation along the backbone. It is worth noting that both Cartesian components of the pressure gradient reach a positive maximum; namely $(\frac{p_x}{\rho g})_{max} \approx 0.47$ and $(\frac{p_y}{\rho g})_{max} \approx 0.06$ for the present simulation. The vertical pressure gradient (being mainly negative) must necessarily change sign before reaching this positive maximum. This is indicated in Fig. 8 b. At a given instant, this change of sign occurs exactly at the highest vertical point reached by the backbone before coming down towards the tip of the crest (marked with a blue circle in Fig. 7). At these points, the fluid particles are hence vertically in free fall. However, the horizontal pressure gradient is not nil, and its maximum non-dimensional horizontal pressure gradient is about $1/2$. Since the wave propagates in the negative x-axis, the fluid particles decelerate horizontally. Finally, at the latest stages of the flow, when the plunging breaker is well developed, the pressure gradient drops to zero as expected since the fluid is in free fall.
5. Focused wave and multiple backbone lines

Since the present potential flow solver cannot reproduce the wave generation by means
of a wavemaker, the initial conditions for this application case are a wave train yielded by
a Boussinesq model (see Kimmoun and Scolan, 2009). A parametric analysis is achieved
in Scolan and Etienne (2023) by varying a multiplicative factor applied to the initial free
surface elevation of the wave train so that the resulting focused wave can break or not.
For this simulation, the errors in mass and energy conservation are at worst 10⁻⁶ and
10⁻⁵ respectively. We analyse here in more detail the simulation obtained with a factor
that yields the limiting case just before breaking. The wave train hence propagates and
focuses up to a very near breaking wave.

In this simulation we choose four instants at which all the backbone lines are identified
in the fluid. Figs. 9, 10, 11 and 12 show the corresponding backbone lines at instants
$t = 0.1$s, $t = 1.1$s, $t = 2.08$s and $t = 2.6$s respectively. The third instant corresponds to
the highest crest over the whole simulation.

We are not interested in the reflection of the wave train at the right wall. Therefore,
the simulation lasts until the main crest slows down significantly and when this crest
approaches the right wall. The bunch of small backbone lines at the left and right walls
could be numerical artifacts. No effort is made to better analyze the pressure in their
vicinity. As the supports of the free surface elevation and tangential velocity at the free
surface are both compact (see Figs. 9 to 11), these spurious effects could be pushed away
far from the wave train of interest; however, at a higher computational cost.

It is worth noting that the backbone lines often emanate from a local maximum of the
wave, especially those backbone lines which reach the bottom. That is why, in the four
plots, the spatial variation of the nondimensional horizontal acceleration $\frac{1}{g} \frac{d\vec{u}}{dt} \cdot \vec{x}$ at the free
surface is drawn. This quantity is also the non dimensional horizontal pressure gradient.
The change of sign of this acceleration component and the relative importance of the
deceleration on the left side of the crest influences how the backbone line emanates from
the bottom. These contact points correspond exactly to the local pressure maxima along
the bottom. This is confirmed by plotting the spatial variation of the non-dimensional
horizontal pressure gradient $|p_{,x}|/(\rho g)$ along the bottom. The position of its zeroes corre-
sponds exactly to the connection of the backbone lines with the bottom. We can define
an averaged orientation of the backbone line by assessing the slope of the straight line
that approximates the backbone line from the bottom to the free surface. This orientation
depends on the acceleration of the crest. Before reaching the maximum velocity (it occurs
roughly at $t = 2.08$s when the maximum crest is reached), the backbone line has an aver-
age positive slope and the crest accelerates. Then the crest slows down, the orientation is
inverted and the backbone line reaches the bottom at a point ahead the crest. It is worth
noting that backbone lines move upstream back to the left wall following the small crests
in the tail of the main wave train of the focused wave.

The backbone lines reach the bottom as long as the associated crests in the wave train
are high enough. However, some backbone lines do not reach the bottom and end inside
the fluid. This is the case for the backbone line enclosed in a rectangular blue frame in
the plot at time $t = 2.6$s. Fig. 13 provides a closer view of the two normalized vector
fields $\frac{\nabla p}{||\nabla p||}$ and $\frac{\vec{e}_1}{||\vec{e}_1||}$ in the vicinity of this small backbone line. In this studied region, the
pressure is mainly hydrostatic, and the vector field of the pressure gradient is directed
downwards and is almost constant. This is not true everywhere, especially on the left part
of the plot, where the vector field of $\frac{\vec{e}_1}{||\vec{e}_1||}$ varies significantly there. In particular, the end point of the backbone line is embedded in a region where the Gaussian curvature of the pressure (determinant of the Hessian matrix) is positive. This means that the curvature radii of the pressure are both negative is this small region. In other words the pressure surface is almost flat in this small region; its tangent plan has a slope $-g$.

It is also worth noting that the horizontal component of $\vec{e}_1$ changes sign abruptly along a line that joins the free surface to the region where the Gaussian curvature is positive. This means that along this line the cross second derivative of the pressure $p_{,xy}$ vanishes. The local analysis of the vertical pressure gradient also shows a saddle point in the close vicinity of the boundary where the Gaussian curvature vanishes (green dots in Fig. 13).

In the results above, the backbone lines are identified in three subdomains to capture all of them with much precision. The tracking of some backbone lines is illustrated in the supplementary material with a video (multi-backbone-focused-wave.avi) albeit with less accuracy. However, the backbone line attached to the main crest (the focused wave which almost breaks) is well captured and tracked.

## 6. Conclusion

The analysis of the spatial variations of the pressure is performed in highly nonlinear waves for four different configurations. In each case, it is shown that tracking over time and space the lines where the pressure gradient is colinear to one of the two eigenvectors of the pressure Hessian matrix reveals new features of the fluid dynamics beneath the free surface and down to the bottom. This line is called backbone line. Its existence is formally shown for analytical solutions that mimic the tip of an overturning crest. Its computation for more realistic configurations allows describing, in a continuous way, the temporal and spatial evolution of the pressure gradient (alternatively the Lagrangian acceleration) long before reaching the Stokes limit and up to the last stage of a plunging breaker.

The pressure maxima (if any) are necessarily located on the backbone line. Tracking this line as it approaches the free surface hence allows for a better description of the onset of a critical jet (cusp) at the free surface. A conjecture arises from this analysis : the tip of an overturning crest turns on the side where the Lagrangian acceleration is the largest. This is true for a critical jet triggered by the occurrence of a local pressure maximum close to the free surface.

The behavior of a wave train leading to a focused wave is strongly correlated with the spatial variation of the pressure gradient at the bottom. As a consequence, the acceleration or slowdown of the main crest is correlated to the temporal and spatial variation of the backbone line at the bottom.

Experimentally it is questionable whether a Particle Image Velocimetry technique could yield the Lagrangian acceleration field with enough accuracy, but this would be a decisive step forward to set physical evidence of the results presented here. At least, experiments should quantify the effects of surface tension and viscosity when highly accelerated flow occur in critical jets.

In future works, the flip-through phenomenon will be re-examined in the light of the pressure gradient analysis proposed here. In particular, tracking its zeros will enable us to better define the timing of maximum fluid acceleration and maximum pressure along the wall. It would be also interesting to examine the three-dimensional extension of the backbone lines.
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<figure><img src="image_6.png" /><figcaption>Figure 4: (a) : free surface profiles at the first time of the restarted simulation (green) and at the last time of the simulation (red). Black lines: lines along which $e_x p_y - e_y p_x = 0$. Subplot of fig (a) : close-up overlapping the main tip and the critical jet.(b) Green vector field: normalized pressure gradient bounded by the blue frame in the subplot in fig (a). Red circle: Lagrangian markers of the free surface. Blue circle: maximum of the Lagrangian acceleration over the entire free surface. The horizontal and vertical axis correspond to the Cartesian coordinates in metre.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 5: (a) : Free surface profile (red). Backbone line (black) described with an arclength starting from the black bullet. The horizontal and vertical axis correspond to the Cartesian coordinates in metre.
(b) Spatial variation of the two Cartesian components of the non-dimensional pressure gradient $(\nabla p/\rho g)$ along the backbone line starting from the black bullet.</figcaption></figure>
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Figure 6: Free surface profiles selected for the analysis of the backbone line. Black line: free surface profile corresponding to the first occurrence of the vertical front. Time step between each profile: 0.01s. The horizontal and vertical axis correspond to the Cartesian coordinates in metre.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 7: (a) : Free surface profiles at initial time of the simulation (green) and at the last computed time (red). Backbone line in black. Subplot of fig (a) : close-up overlapping the main tip.(b) : Green vectors : vector field of the normalized pressure gradient bounded by the blue frame of the subplot in fig (a). Red circle: Lagrangian markers of the free surface. Blue circle: maximum altitude of the backbone line where the vertical pressure gradient also changes sign. The horizontal and vertical axis correspond to the Cartesian coordinates in metre.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>Figure 8: Temporal and spatial variation of the Cartesian components of the pressure gradient along the backbone lines associated to the free surface profiles shown in Fig. 6. The origin of the arclength correspond to the vertical coordinate $y = 0.9891m$ in Fig. 6. (a) : Temporal and spatial variation of $\frac{p_{,x}}{\rho g}$, Light blue dots: maximum of $\frac{p_{,x}}{\rho g}$. (b) : Temporal and spatial variation of $\frac{p_{,y}}{\rho g}$, Light blue dots: zeroes of $\frac{p_{,y}}{\rho g}$. (c) : Temporal and spatial variation of $\frac{p_{,y}}{\rho g}$, close-up about the threshold $\frac{p_{,y}}{\rho g} = -1/2$. Black line : $\frac{p_{,y}}{\rho g}$ along the backbone line when the free surface has a vertical front. Blue dots : points along the backbone line where $\frac{p_{,y}}{\rho g}$ has 2 extrema: minimum and maximum. Green dots : points along the backbone line where $\frac{p_{,y}}{\rho g}$ passes the threshold $-1/2$.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure 9: Red : free surface profile at instant $t = 0.1s$. The left vertical and horizontal axes indicate the Cartesian coordinates of a point in metre. Black: backbone lines. Green : non dimensional horizontal acceleration $\frac{1}{g}\frac{d\vec{v}}{dt}\cdot\vec{x}$ along the free surface measured with the right vertical axis. Magenta : non dimensional horizontal pressure gradient $|p_{,x}|/(\rho g)$ along the bottom measured with the left vertical axis.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 10: Free surface profile at instant $t = 1.1s$. For explanation see caption of Fig. 9.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Figure 11: Free surface profile at instant $t = 2.08s$. For explanation see caption of Fig. 9.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Figure 12: Free surface profile at instant $t = 2.6s$. For explanation see caption of Fig. 9. The blue rectangular frame corresponds to the closer view of the vector fields plotted in Fig. 13.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure 13: Vector fields of the normalized vector fields $\frac{\vec{\nabla}p}{||\nabla p||}$ (cyan) and $\frac{e_1}{||e_1||}$ (blue). Black: backbone line. Red: free surface profile. Green: closed region where the Gaussian curvature of the pressure is positive. The horizontal and vertical axis correspond to the Cartesian coordinates in metre.</figcaption></figure>
**Declaration of interests**

[x] The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

[ ] The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: