Extreme internal waves: gravity currents and overturning fronts

TL;DR

The paper advances the rigorous understanding of extreme internal waves in a two-layer, free-boundary Euler setting by analyzing two large-amplitude bore families: ${\mathscr C}_{\textup{elev}}$ (elevation) and ${\mathscr C}_{\textup{depr}}$ (depression). It proves that elevation bores necessarily overturn, while depression bores either overturn or converge to gravity currents as the front approaches a wall; in the Boussinesq limit, gravity currents satisfy a sharp $60^{\circ}$ wall-contact angle, confirming von Kármán’s conjecture in this regime. The authors develop novel geometric-analytic tools—domain-variational formulations, energy-density bounds, and Monotonicity formulas of Alt–Caffarelli–Friedman and Varvaruca–Weiss type—to perform blow-up analyses and classify limiting configurations, including gravity-current limits and their contact angles. These techniques provide a robust framework for analyzing large-amplitude, front-type waves in multi-fluid free-boundary problems and have potential applications to broader bifurcation studies of high-impact wave phenomena. The results yield rigorous confirmation of longstanding numerical observations and enrich the mathematical theory of overturning and gravity-current limits in stratified fluids, with implications for oceanic and engineering contexts.

Abstract

Hydrodynamic bores are front-type traveling wave solutions to the two-layer free boundary Euler equations in two dimensions. The velocity field in each layer is assumed to be incompressible and irrotational, and it limits to distinct laminar flows upstream and downstream. Rigid horizontal boundaries confine the fluids from above and below. A constant gravitational force acts on the waves, but surface tension is neglected. It was recently shown by the authors that there exist two large-amplitude families of hydrodynamic bores: a curve of depression bores and a curve of elevation bores. We now prove that in the limit along the elevation bore family, the solutions must overturn: the interface separating the layers develops a vertical tangent. This type of behavior was first observed over 45 years ago in numerical computations of internal gravity waves and gravity water waves with vorticity. Despite considerable progress over the past decade in constructing families of water waves that potentially overturn, a proof that overturning definitively occurs has been stubbornly elusive. We further show that in the limit along the depression bore family, either overturning occurs or the solutions converge to a gravity current: the free boundary contacts the upper wall and the relative velocity in the upper fluid is stagnant. We also determine the contact angle between the interface and the rigid barrier for the limiting gravity current, giving the first rigorous confirmation of a conjecture of von Kármán. The resolutions of these questions in the specific case of hydrodynamic bores is accomplished through the use of novel geometric analysis techniques, including bounds on the decay of the velocity field near a hypothetical double stagnation point. These ideas may have broader applications to bifurcation theoretic studies of large-amplitude waves.

Figures (9)

• Figure 1: A steady hydrodynamic bore viewed in a frame moving with the front. The channel extends from $y=-h_1$ to $y = h_2$, with the heavier fluid layer (shaded in darker blue) $\mathscr{D}_2$ lying below the lighter fluid layer $\mathscr{D}_1$. Both upstream and downstream, the limiting velocity field is purely horizontal. Note that the upstream velocity is the same in both layers, here normalized to $(-1,0)$, whereas there will be a jump in the downstream velocity.
• Figure 2: (a) The gravity current model proposed by von Kármán vonkarman1940engineer. A heavier fluid (shaded darker blue) intrudes on a lighter fluid (shaded light blue). The velocity in the heavy fluid is assumed to be constant. In the moving frame, the heavy fluid appears to be at rest, while the lighter fluid flows over it. The free boundary meets the bed at a stagnation point (in both phases). (b) Benjamin's benjamin1968gravity cavity front model. There is a single phase, which is emptying out of the channel far downstream.
• Figure 3: Limiting configurations observed numerically by Dias and Vanden-Broeck dias2003internal and verified in Theorems \ref{['overturning theorem']}--\ref{['non-boussinesq gravity current theorem']} in the non-Boussinesq setting. (a) Bores of elevation ${\mathscr C}_{\textup{elev}}$ invariably overturn. Along the curve of depression bores ${\mathscr C}_{\textup{depr}}$, either (b) overturning occurs or (c) the free boundary limits to the upper wall, resulting in a gravity current where the free boundary meets the wall tangentially.
• Figure 4: Limiting configurations in Theorem \ref{['boussinesq gravity current theorem']} in the Boussinesq setting. Along the curve of elevation bores ${\mathscr C}_{\textup{elev}}$, either (a) overturning occurs or (b) the free boundary limits to the upper wall, resulting in a gravity current where the free boundary makes a contact angle of exactly $60\degree$. The alternatives (c) and (d) for depression bores ${\mathscr C}_{\textup{depr}}$ are similar. Dias and Vanden-Broeck dias2003internal numerically observe (b) and (d) but not (a) or (c).
• Figure 5: Possible blowup limits $u_0$ in a neighborhood of the origin. The components of $\{u_0 < 0\}$ are shaded in darker blue, components of $\{u_0 > 0\}$ are shaded in lighter blue, $\operatorname{int}\{ u =0\}$ is left white, and $\Gamma(u_0)$ is indicated by the (non-dashed) lines. Both (a) and (b) correspond to \ref{['three component single phase blowup u']}; note that any rotation of these two configurations is also possible. Cases (c)--(h) are captured by the ansatz \ref{['three component two phase blowup u']}, while (i)--(k) correspond to \ref{['two Stokes flows']}. For monotone solutions, only the vanishing limit of Case (l) can occur.

Theorems & Definitions (58)

• Theorem 1.1: Overturning
• Theorem 1.2: Gravity current or overturning
• Theorem 1.3: Limiting Boussinesq bores
• Remark 1.4
• Theorem 2.1: ACF monotonicity formula
• Theorem 2.2: Oddson
• Theorem 2.3: Large-amplitude bores
• Theorem 2.4
• proof
• Definition 3.1