Stokes Waves with Piecewise Smooth Vorticity

• The paper introduces a framework that generalizes classical Stokes waves by incorporating piecewise smooth vorticity, revealing new bifurcation and dispersion properties.
• It employs hodograph transformations and Sturm–Liouville analyses to handle transmission problems across internal shear layers.
• Numerical methods confirm large-amplitude waves with overhanging profiles and critical layers, providing practical insights into complex shear dynamics.

A Stokes wave with piecewise smooth vorticity is a periodic gravity wave propagating along the free surface of a two-dimensional, inviscid, incompressible fluid, where the underlying vorticity of the flow is a function with discontinuities (jumps) at finitely many streamlines. This generalizes the classical irrotational and constant-vorticity Stokes waves by incorporating internal layers of differing vorticity, thus modeling shear flows with abrupt transitions—physically relevant in geophysical and engineering contexts where wind, stratification, or boundary shear create layers with distinct vorticity profiles.

1. Formulation and Transmission Structure

Consider a two-dimensional fluid domain of infinite or finite depth, with coordinates $(x, y)$ and a free surface given by $y = \eta(x)$, periodic in %%%%2%%%% with period $2\pi$. The fluid occupies $\Omega_\eta=\{(x, y): y<\eta(x)\}$, and the wave travels steadily at speed $c$. The flow is described by the stream function $\psi(x,y)$, such that $(u, v) = (\psi_y, -\psi_x)$, satisfying incompressibility and the steady Euler equations. The vorticity $\omega(x,y)=\partial_x v-\partial_y u$ is prescribed as a piecewise $C^1$ function: $$\omega(x, y) = \begin{cases} \omega_1(y), & y > y_0, \ \omega_2(y), & y < y_0, \end{cases}$$ where $y_0$ is a horizontal interface at which the vorticity jumps. The key physical boundary conditions are:

• Kinematic: $v = (u - c)\eta_x$ on $y = \eta(x)$.
• Dynamic: pressure $P$ constant on $y = \eta(x)$.
• Decay: $(u, v) \to (0, 0)$ as $y \to -\infty$ (in deep water).
• Transmission: $u$, $v$, and $P$ are continuous across $y = y_0$, while $\omega$ has a jump.

The governing steady equations reduce, via the stream function, to: $-\Delta \psi(x, y) = \omega(\psi(x,y))$ plus matching and Bernoulli conditions at the boundaries and the internal interface.

2. Analytical Framework: Height Function and Hodograph Methods

To analyze these free boundary problems, a hodograph transformation is commonly employed. Here, the change of variables $q = x$, $p = -\psi(x, y)$ maps the physical domain to a strip $$D = \{ (q, p): q \in \mathbb{R}/2\pi\mathbb{Z},\, 0 < p < \infty \}$$, with the free surface $p = 0$ and the internal interface at $p = p_0$ corresponding to $y = y_0$. The height function $h(q, p)$ gives $y$ as a function of $(q, p)$. The transformed PDE for $h$ is quasilinear elliptic: $$(1 + h_q^2)h_{pp} - 2 h_q h_p h_{qp} + h_p^2 h_{qq} + \gamma(-p) h_p^3 = 0,$$ where $\gamma$ is the vorticity profile in terms of streamlines. The boundary and transmission conditions become:

• Bernoulli at $p=0$: $1 + h_q^2 + 2g h h_p^2 = 0$.
• Far-field: $(h_q, h_p) \to (0, 1/c)$ as $p \to \infty$.
• Transmission at $p=p_0$: Continuity of $h$ and $h_p$.

When $\gamma$ is only piecewise $C^1$, $h$ is defined separately in the upper and lower layers, matched at $p = p_0$ with transmission conditions. The presence of internal interfaces leads to a transmission problem, structurally more complex than the classical single-layer setting (Gui et al., 6 Nov 2025).

3. Bifurcation and Dispersion Analysis

The existence and local bifurcation of small-amplitude Stokes waves are governed by dispersion equations. For finite depth and piecewise smooth vorticity, the general approach is:

• Compute a background parallel shear flow $U(Y)$ in $0 \leq Y \leq h$ by solving $U''(Y) + \omega(U) = 0$, imposing $U(0) = 0$, $U(h) = 1$, and a Bernoulli-type condition for the surface slope.
• For each wavenumber parameter $\tau > 0$, solve the piecewise Sturm–Liouville problem: $$-\gamma''(Y) + [\tau^2 - \omega'(U(Y))]\gamma(Y) = 0,$$ with matching conditions at jumps in $\omega'$, and Dirichlet boundary conditions. Continuity of $\gamma$ and $\gamma'$ is required at the interfaces.
• The small Stokes waves bifurcate when the dispersion determinant $D(\tau)$ vanishes; this generalizes the familiar constant-vorticity dispersion law.
• Two bifurcation scenarios arise: fixed Bernoulli constant or fixed wavelength. The former tracks amplitude branches as Bernoulli parameter $r$ varies; the latter treats $r$ (or an auxiliary parameter) as a function of fixed wavenumber (Kozlov et al., 2012).

In the piecewise setting, the solvability conditions and spectrum are determined by the transmission-formulated Sturm–Liouville problem, leading to real roots corresponding to bifurcation points. Sufficient conditions are described in terms of the sign of ratios like $R'/h'$ and the Dirichlet eigenvalue count in the subdomains. For cases with $k$ Dirichlet eigenvalues, there will be $k+1$ positive real roots under appropriate monotonicity (Kozlov et al., 2012).

4. Numerical Approaches for Large-Amplitude and Complex Stokes Waves

Analytical treatment beyond small amplitude is challenging due to loss of global compactness and non-Fredholmness (in deep water). Advanced numerical schemes are therefore utilized for computation of large-amplitude waves, critical layers, and profiles with internal jumps:

• The conformal mapping method transforms the fluid domain to a fixed rectangle parameterized by $(\alpha, \beta)$, where elliptic PDEs for the streamfunction $\psi$ and the conformal mapping functions $(X, Y)$ are discretized using finite differences.
• Boundary conditions are imposed on the mapped rectangle: Dirichlet for the bottom, periodicity in the horizontal variable, and nonlinear Bernoulli condition at the moving free surface.
• For piecewise-constant or piecewise-smooth vorticity, the internal interface is identified by the value of $\psi = \psi_c$, and matching conditions on $\psi$ and the normal velocity are enforced at this curve by suitable interpolation.
• The resulting nonlinear system of equations in discrete variables $\{\psi_{m, n}, Y_{m, n}\}$ is solved via Newton–Raphson iteration, with continuation in amplitude or vorticity parameters (Doak et al., 25 Feb 2025).

This approach accommodates internal stagnation points, critical layers, and overhanging free-surface profiles, including cases where the vorticity jumps lead to complex modal structure and resonance phenomena between layers.

5. Main Existence Theorems and Bifurcation Outcomes

Rigorous existence results for large-amplitude Stokes waves with piecewise smooth vorticity in infinite-depth domains employ bifurcation-theoretic and topological methods:

• The linearized operator about a flat laminar shear flow is typically non-Fredholm in deep water due to lack of compactness. Introducing a vanishingly small artificial damping renders the operator Fredholm, permitting the application of analytic (Crandall–Rabinowitz/Dancer/Buffoni–Toland) bifurcation theory.
• For piecewise $C^{1,\alpha}$ vorticity vanishing at depth and with suitable decay, there exists a global connected bifurcation branch of nontrivial periodic solutions. Along this branch, the solutions either attain arbitrarily large wave speed or approach horizontal stagnation where the horizontal velocity approaches the phase speed ($u \to c$), corresponding to a possible limiting wave of greatest height (Gui et al., 6 Nov 2025).
• The persistence of “nodal pattern” (monotonicity properties of $h_q$ and $h_{qq}$) implies no overhanging in constructed global branches under current proof techniques, though numerical evidence demonstrates overhanging is possible for finite amplitude at certain vorticities (Doak et al., 25 Feb 2025).

6. Physical Implications, Model Examples, and Bifurcation Structure

Piecewise smooth vorticity supports a richer variety of dynamics than single-layer or constant-vorticity settings:

• Internal shear layers (vorticity jumps) induce new modes, critical layers, and stagnation regions (“bubbles”) in the interior. A sign change in vorticity results in broad critical layers rather than isolated stagnation points.
• Overhanging free-surface profiles appear for sufficiently strong upper-layer vorticity, particularly when $|\omega_1-\omega_2|$ is large. The layer thickness ratio and vorticity contrast control the modal structure, with bifurcations where new overhanging branches merge or emerge.
• In the small-amplitude regime, explicit dispersion relations exist for two-layer constant and linear vorticity profiles, generalizing classical formulas (Kozlov et al., 2012). At larger amplitude, these bifurcations produce families of large Stokes waves with observed properties:
  • Multiple dispersion branches and resonant mode interactions.
  • Extreme profiles with free-surface overhangs or trapped stagnant regions at internal layers.
  • Consistency with analytical predictions in the constant/linear/irrotational limits (Doak et al., 25 Feb 2025, Gui et al., 6 Nov 2025).

The deterministic framework summarized above establishes the bifurcation and persistence of Stokes waves with piecewise smooth vorticity, with validation across analytic, spectral, and direct numerical domains. The transition from irrotational and single-vorticity regimes to layered, discontinuous vorticity settings uncovers substantially richer mathematical structure and physical behavior, relevant for engineering flows, oceanography, and the theoretical aspects of water wave bifurcation theory.