On the vorticity threshold for steady water waves

Problem Formulation
This paper investigates two-dimensional steady water waves in a fluid of constant density and finite depth, governed by the full steady Euler equations. The study focuses on rotational flows with constant, adverse (negative) vorticity, denoted by $\omega = -\gamma < 0$. The fluid domain is bounded below by a flat bed and above by a free surface $y = \eta(x)$. The authors restrict their analysis to "unidirectional" waves, where the horizontal relative velocity $u$ is strictly positive everywhere in the fluid, excluding the formation of internal recirculation zones (eddies) within the bulk flow. The study considers both periodic Stokes waves and solitary waves, assuming smooth classical solutions.

The central physical question addressed is the behavior of these waves as the magnitude of the adverse vorticity $\gamma$ increases. While numerical observations suggest that strong adverse vorticity can lead to extreme wave heights and surface singularities (similar to irrotational waves), other numerical studies indicate a transition where stagnation points first appear on the bottom boundary rather than the surface. This paper seeks to rigorously establish the conditions under which bottom stagnation occurs and to characterize the wave properties in the regime of large adverse vorticity.

Methodology
The authors employ a stream function formulation of the steady Euler equations, where the vorticity is constant. The problem is reduced to an elliptic boundary value problem for the stream function $\psi$, subject to kinematic and dynamic boundary conditions on the free surface and the bed.

The core of the analysis relies on the maximum principle and the construction of auxiliary functions to derive pointwise estimates for the velocity field. Specifically:

1. Auxiliary Functions: The authors introduce functions of the form $f = \frac{1}{2}|\nabla \psi|^2 + y + (\lambda - \gamma)\psi$ and $h = \psi_y - \lambda y$ to analyze the behavior of the vertical derivative of the stream function (which corresponds to the horizontal velocity $u$).
2. Maximum Principle: By demonstrating that these auxiliary functions satisfy specific elliptic inequalities (e.g., $\Delta f + a f_x + b f_y \geq 0$), the authors use the maximum principle and the Hopf lemma to determine where extrema of the velocity occur.
3. Laminar Flow Analysis: The study utilizes explicit formulas for laminar (trivial) shear flows to establish baseline relationships between the Bernoulli constant $r$, the vorticity $\gamma$, and the flow depth. This provides a reference for bifurcation analysis.
4. Bifurcation Theory: The results are framed within the context of global bifurcation curves of solutions connected to laminar flows, utilizing existing global existence results for solitary and Stokes waves.

Key Contributions and Results

• Existence of a Critical Vorticity Threshold: The primary theoretical contribution is the identification of an explicit critical threshold for adverse vorticity, $\gamma_\star \approx 1.37184$. The authors prove that for any constant adverse vorticity $\gamma > \gamma_\star$, a specific lower bound on the surface velocity holds.
• Bottom Stagnation Precedence: For $\gamma > \gamma_\star$, the paper proves that the first stagnation point (where $u=0$) in a continuous family of unidirectional waves must appear on the bottom boundary directly beneath the wave crest. This is established by showing that the horizontal velocity $u$ on the surface satisfies the lower bound $u \geq \frac{1}{2}\gamma \eta$. Consequently, extreme waves with surface singularities or overhanging profiles cannot form before a bottom stagnation point emerges.
• Existence of Solitary Waves with Bottom Stagnation: Combining the threshold result with global bifurcation theory, the authors prove the existence of smooth solitary waves that possess exactly one stagnation point located at the bottom beneath the crest, while maintaining unidirectionality ($u>0$) everywhere else in the fluid.
• Asymptotic Estimates for Large Vorticity: For large $\gamma$, the paper derives sharper asymptotic estimates for solitary waves bifurcating from shear flow:
  • The surface velocity at the crest satisfies $u \geq \sqrt{2\gamma} - O(\gamma^{-5/2})$.
  • The maximum wave amplitude (depth at the crest) is bounded by $\hat{\eta} \leq \sqrt{2/\gamma} + O(\gamma^{-7/2})$.
  • The wave amplitude (difference between crest and trough) scales as $O(\gamma^{-2})$.
  • Similar bounds are established for Stokes waves provided the Bernoulli constant satisfies a specific upper bound.

Significance and Claims
The paper claims to resolve a discrepancy between numerical observations and theoretical expectations regarding the damping effect of strong adverse vorticity. While some numerical studies predicted that strong adverse currents would increase wave height and steepness, this work demonstrates that under the strict assumption of unidirectionality, strong adverse vorticity actually imposes a damping effect on the amplitude, limiting the wave height.

The authors argue that the discrepancy with numerical predictions of increasing wave height arises because such large-amplitude waves in strong adverse currents likely violate the unidirectionality assumption, leading to the formation of critical layers and eddies. By establishing the threshold $\gamma_\star$, the paper delineates the regime where unidirectional solutions are constrained to have bottom stagnation before any surface singularities can occur. This provides a rigorous mathematical explanation for the transition from surface-dominated stagnation (in low vorticity or irrotational cases) to bottom-dominated stagnation in high adverse vorticity regimes.

The work is presented as a theoretical and mathematical analysis; the authors explicitly state that no datasets were generated or analyzed, and the results are derived entirely from the properties of the Euler equations and elliptic PDE theory.