1. A theoretical framework for irrotational breaking waves
• An extension of the Zakharov-Craig-Sulem formulation for parametrised surfaces
• Hamiltonian structure and the link with the Water Waves equations
2. Did you just say "irrotational"?
• Numerical method for the Navier-Stokes system
• When are water waves really irrotational?

In Craig (2017), the 2D irrotational Euler equation is rephrased as a 1D equation (\(d=1\)) using quantities defined only on the free surface.

What we would like to do:

• Investigate the Hamiltonian structure
• Find a link with the Water Waves equations of Zakharov (1968)Craig & Sulem (1993)
• Find a formulation with \(d=2\)

Other formulations encompassing wave breaking:

• Sulem, Sulem, Bardos & Frisch (1981)
• Benjamin & Olver (1982)
• Wu (1997, 1999)

We now assume that \(\boldsymbol{\omega} = \boldsymbol{\nabla}\times \boldsymbol{u} = 0\)

Hence, \( \boldsymbol{u} = \boldsymbol{\nabla} \phi \) for some potential \(\phi\)

Introduce the free-surface potential \(\psi\) \[ \psi(t,\vec{s}) = \phi\big( t, \boldsymbol{\gamma}(t,\vec{s}) \big) \]

Euler's system, \[\begin{align} \partial_t \boldsymbol{u} + \boldsymbol{u}\cdot \boldsymbol{\nabla}\boldsymbol{u} &= - \boldsymbol{\nabla} p + \boldsymbol{g} \\ \boldsymbol{\nabla} \cdot \boldsymbol{u} &= 0 \end{align}\] Assuming irrotationality, we obtain (Daniel) Bernoulli's law in the Eulerian frame, \[\partial_t \phi \style{color: #42affa;}{+} \frac{1}{2}\, \big| \boldsymbol{\nabla} \phi \big|^2 + p + g z = 0 \] On the free surface, this becomes \[\partial_t \phi \big|^2\Big|_{\boldsymbol{\gamma}(t,\vec{s})} + \frac{1}{2}\, \big| \boldsymbol{\nabla} \phi \big|^2\Big|_{\boldsymbol{\gamma}(t,\vec{s})} + g \gamma_z = 0 \]

Tangent vectors: \[\hat{\boldsymbol{\tau}}_i = \frac{\partial_{s_i} \boldsymbol{\gamma}}{|\partial_{s_i} \boldsymbol{\gamma}|}\] Intrinsic metric tensor: \[\mathsf{g}_{ij} = \begin{bmatrix} \partial_{s_i} \boldsymbol{\gamma} \cdot \partial_{s_j} \boldsymbol{\gamma} \end{bmatrix}\]

\[ \left\{ \begin{array}{rcl} \partial_t \boldsymbol{\gamma} &=& \cfrac{\mathrm{DtN}[\boldsymbol{\gamma}]\psi}{\sqrt{\det \mathsf{g}}} \,\hat{\boldsymbol{n}} + \begin{bmatrix} \partial_{s_1} \psi & \partial_{s_2} \psi \end{bmatrix} \mathsf{g}^{-1} \begin{bmatrix} \partial_{s_1} \boldsymbol{\gamma} \\ \partial_{s_2} \boldsymbol{\gamma} \end{bmatrix} \\ \partial_t \psi &=& -g \gamma_z + \cfrac{1}{2} \cfrac{\big(\mathrm{DtN}[\boldsymbol{\gamma}]\psi\big)^2}{\det \mathsf{g}}\style{color: #42affa;}{+} \begin{bmatrix} \partial_{s_1} \psi & \partial_{s_2} \psi \end{bmatrix} \mathsf{g}^{-1} \begin{bmatrix} \partial_{s_1} \psi \\ \partial_{s_2} \psi \end{bmatrix} \end{array} \right. \]

The system of Zakharov (1968)Craig & Sulem (1993) \[ \left\{ \begin{array}{rcl} \partial_t h &=& \mathrm{DtN}[h]\Psi \\ \partial_t \Psi &=& -g h - \cfrac{1}{2} \big|\vec{\nabla}\Psi\big|^2 + \cfrac{1}{2} \cfrac{\big( \mathrm{DtN}[h]\Psi + \vec{\nabla}\Psi \cdot \vec{\nabla} h \big)^2}{1 + |\vec{\nabla} h|^2} \end{array} \right. \]

Defining the following Hamiltonian functional, \[ \mathrm{H}[h,\Psi] = \frac{1}{2} \int_{\mathbb{R}^d} \Psi\ \mathrm{DtN}[\boldsymbol{\gamma}]\Psi + \frac{g}{2} \int_{\mathbb{R}^d} h^2 \] the Water Waves equations can be recast as \[ \partial_t \begin{bmatrix} h \\ \Psi \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} \delta_h \mathrm{H} \\ \delta_\Psi \mathrm{H} \end{bmatrix} \]

Define the following Hamiltonian functional

\[ \mathrm{H}[\boldsymbol{\gamma},\psi] = \frac{1}{2} \int_{\mathbb{R}^d} \psi\ \mathrm{DtN}[\boldsymbol{\gamma}]\psi + \frac{g}{2} \int_{\mathbb{R}^d} \big(\gamma_z\big)^2\, \hat{\boldsymbol{z}} \cdot \hat{\boldsymbol{n}}\, \sqrt{\det \mathsf{g}} \]

then, we can show (when \(d=1\) or \(2\)) that \[ \left\{ \begin{array}{rccccl} \partial_t \boldsymbol{\gamma} &=& & \cfrac{\boldsymbol{n}}{\det \mathsf{g}} && \cfrac{\delta \mathrm{H}}{\delta \psi} + \text{tangential terms} \\ \partial_t \psi &=& -& \cfrac{\boldsymbol{n}}{\det \mathsf{g}} & \cdot & \cfrac{\delta \mathrm{H}}{\delta \boldsymbol{\gamma}} + \text{tangential terms} \end{array} \right. \] This is in fact a Hamiltonian system with non-canonical symplectic structure (following the definitions ofBenjamin & Olver (1982) or Kuksin (2000)).

1. A theoretical framework for irrotational breaking waves
• An extension of the Zakharov-Craig-Sulem formulation for parametrised surfaces
• Hamiltonian structure and the link with the Water Waves equations
2. Did you just say "irrotational"?
• Numerical method for the Navier-Stokes system
• When are water waves really irrotational?

We use the first order (in \(a\)) solution (i.e. a linear wave) of amplitude \(a = 0.5\), \[ \boldsymbol{\gamma}(t=0,s) = \begin{bmatrix} s \\ h_0 + a \cos(kx) \end{bmatrix}\qquad\text{and}\qquad \boldsymbol{u}_0(x,z) = \frac{a\omega}{\sinh(kh_0)} \begin{bmatrix} \cosh(k\style{color: #42affa;}{h_0}) \cos(kx) \\ \sinh(k\style{color: #42affa;}{h_0}) \sin(kx) \end{bmatrix} \] Then we solve numerically the following elliptic problem,

At last, the initial velocity is computed as \( \boldsymbol{u}(t = 0) = \boldsymbol{\nabla} \phi_0 \)

We use the FreeFEM library Hecht (2012) to

• generate and handle the mesh
• build and handle the matrices
• use the PETSc methods easily

Figure - The mesh generated from the initial free surface (\(\approx 200\,000\) triangles).

Figures (from R. & Dormy (2024)) - Convergence to the inviscid solution (computed using the numerical method of Dormy & Lacave (2024) )

A local equation for the kinetic energy can be obtained directly from the Navier-Stokes system, \[ \frac{\mathrm{D}}{\mathrm{D}t} \left( \frac{\boldsymbol{u}^2}{2} \right) = \boldsymbol{u} \cdot \boldsymbol{g} - \boldsymbol{u}\cdot \boldsymbol{\nabla} p \style{color: #42affa;}{\ - \frac{1}{\mathrm{Re}}\, \Big[ \boldsymbol{\nabla}\cdot \big( \boldsymbol{u}^{\perp} \omega \big) + \omega^2 \Big]} \] with \( \boldsymbol{u}^{\perp} = [-u_y, u_x] \). The dissipation arises in the support of the vorticity \(\omega\).

Generation of vorticity can be understood in light of the stress-free boundary condition, \[ \begin{align} p \hat{\boldsymbol{n}} - \frac{1}{\mathrm{Re}} \, \Big[ \boldsymbol{\nabla} \boldsymbol{u} + \big(\boldsymbol{\nabla}\boldsymbol{u}\big)^{\top} \Big]\, \hat{\boldsymbol{n}} = 0 \qquad & \Longrightarrow \qquad \hat{\boldsymbol{\tau}} \, \Big[ \boldsymbol{\nabla} \boldsymbol{u} + \big(\boldsymbol{\nabla}\boldsymbol{u}\big)^{\top} \Big]\, \hat{\boldsymbol{n}} = 0 \\ & \Longrightarrow \qquad \omega = 2 \kappa \big( \boldsymbol{u} \cdot \hat{\boldsymbol{\tau}} \big) - 2 \partial_s \big( \boldsymbol{u} \cdot \hat{\boldsymbol{n}} \big) \end{align} \]

Stress-free boundary condition on the free surface \[ \begin{align} p \hat{\boldsymbol{n}} - \frac{1}{\mathrm{Re}} \, \Big[ \boldsymbol{\nabla} \boldsymbol{u} + \big(\boldsymbol{\nabla}\boldsymbol{u}\big)^{\top} \Big] \, \hat{\boldsymbol{n}} &= 0 && \text{on } \Gamma_i(t) \end{align} \] Navier No-slip/Dirichlet boundary condition on the bottom topography \(\Gamma_b\), \[ \boldsymbol{u} = 0\qquad\text{on } \Gamma_b \]

Figure (from R. & Dormy (2025, submitted)) - Comparison of the free surfaces in the case of a water wave propagating over a sharp rectangular step, for different values of Reynolds' number \(\mathrm{Re}\). The "Euler" result has been computed using the numerical method of Dormy & Lacave (2024).

We now use the no-slip/Dirichlet boundary conditions \( \boldsymbol{u} = 0 \) on \(\Gamma_b\)

Animated Figure (from R. & Dormy (2025, submitted)) - Interface evolution for a Reynolds number \(\mathrm{Re} = 10^5\), with a non-breaking water wave propagating over a sharp rectangular step.

Phenomenon already observed by Lin & Huang (2010)

We now use the no-slip/Dirichlet boundary conditions \( \boldsymbol{u} = 0 \) on \(\Gamma_b\)

Animated Figure (from R. & Dormy (2025, submitted)) - Interface evolution for a Reynolds number \(\mathrm{Re} = 10^5\), with a non-breaking water wave propagating over a smoothed/mollified rectangular step.

When can we expect a convergence of the viscous solution to the inviscid (irrotational) one as \(\mathrm{Re}\to + \infty\)?

• Without bottom topography:
  ⤷Masmoudi & Rousset (2017) (with small amplitude assumption)
  ⤷A similar result should hold up to the splash singularity
• Topography \(\Gamma_b\) on which the Navier boundary condition is enforced:
  ⤷Evidences that it should work over a flat topography (up to the splash singularity)
  ⤷Non-flat topographies \(\to\) not tested yet!
• Topography \(\Gamma_b\) on which the no-slip/Dirichlet boundary condition is enforced:
  ⤷Evidences that it should not work over non-flat topographies, the boundary layer might separate.
  ⤷Should the topography be flat, we cannot conclude...

Merci de votre attention!

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We define the function space \[ \mathbf{H}^1_{\Gamma_b}\big(\Omega(t)\big) = \Big\{ \boldsymbol{v} \in H^1 \big( \Omega(t) ; \mathbb{R}^2 \big) : \boldsymbol{v} \cdot \hat{\boldsymbol{v}} = 0 \text{ on } \Gamma_b \Big\} \] without assuming the incompressibility directly in the function space!

Let us consider the Poisson equation, \[-\Delta \psi = \omega\qquad\text{in }\Omega \subset \mathbb{R}^2\]